Riordan Arrays, Elliptic Functions and their Applications

Transcription

1 Riordan Arrays, Elliptic Functions and their Applications by Arnauld Mesinga Mwafise A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the Department of Computing and Mathematics Waterford Institute of Technology Supervisor: Dr Paul Barry Former Head of the School of Science Waterford Institute of Technology June 07

2 Declaration of Authorship I, ARNAULD MESINGA MWAFISE, declare that this thesis titled, Riordan Arrays, Elliptic Functions and their Applications and the work presented in it are my own I confirm that: This work was done wholly or mainly while in candidature for a research degree at this Institute Where I have consulted the published work of others, this is always clearly attributed 3 Where I have quoted from the work of others, the source is always given With the exception of such quotations, this thesis is entirely my own work 4 I have acknowledged all main sources of help Signed: Date:

3 Abstract Riordan arrays have been used as a powerful tool for solving applied algebraic and enumerative combinatorial problems from a number of different settings in pure and applied mathematics This thesis establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays These elliptic Riordan arrays were found in many cases to be useful constructs in generating combinatorially and algebraically significant sequences based on their corresponding trigonometric and hyperbolic forms In addition, in some cases the elliptic Riordan arrays presented interesting structural patterns that were further investigated By exploring elliptic Riordan arrays more closely with respect to other fields, several new applications of Riordan arrays associated with physics and engineering are illustrated Furthermore, other non-elliptic type Riordan arrays having important applications are also presented based on the connection established in the thesis between Riordan arrays and the analytic solutions to some of the families of the Sturm-Liouville differential equations

4 Acknowledgements I would like to express my profound gratitude and appreciation to my supervisor Dr Paul Barry for the excellent research supervision he provided me during my time as a research student I also appreciate all his personal contributions of time and resources to make my doctoral research experience stimulating and productive It has been an honor to be his second PhD student I would also like to thank Dr Aoife Hennessy for serving as my co-supervisor I gratefully acknowledge the funding received to undertake this research project under the Waterford Institute of Technology PhD Scholarship Programme 03

5 Notations and Abbreviations gf Generating Function egf Exponential generating function ogf Ordinary generating functions fps Formal power series Equivalence relation x a(mod b) Congruent relation Multiplication Z The set of Integers N The set of Natural numbers R The set of Real numbers C The set of complex numbers ODE Ordinary Differential Equation PDE Partial Differential Equation [z n ]f(z) The coefficient of z n in the power series f(z) f(z) or Rev(f(z)) the series reversion of the series f(z) R A Riordan array R The Riordan matrix such that R n,k = R n+,k (Axxxx) A number The Online Encyclopedia of Integer Sequences (OEIS) (g, f) An ordinary Riordan array [g, f] An exponential Riordan array x The Floor Function m Elliptic modulus

6 K(m) Complete elliptic integral of the first kind F (φ, m) Elliptic integral of the first kind E(m) Complete elliptic integral of the second kind E(φ, m) Elliptic integral of the second kind Weierstrass elliptic function ζ Zeta pseudo-elliptic function σ Sigma psedo-elliptic function sm Dixonian elliptic sine function cm Dixonian elliptic cosine function sn(z, m) sn(z m) Elliptic sine cn(z, m) cn(z m) Elliptic cosine dn(z, m) dn(z m) Difference function Or st such that For every An element of wrt with respect to 3

7 Contents Introduction to Riordan Arrays and Elliptic Functions Background of the study A brief historical time-line on the development of Riordan arrays 4 3 Definitions and Examples of Riordan Arrays 6 3 Definitions of Riordan arrays 6 3 Some Basic Examples of Riordan arrays 8 4 The Group Structure of Riordan Matrices 0 4 The Sub-groups of the Riordan Group 5 Sequence Characterization of Riordan arrays 5 The Fundamental Theorem of Riordan Arrays 5 The A-Sequence 53 The Z-Sequence 3 6 Production Matrices and Matrix Characterisation of Riordan Arrays 3 6 Production Matrices of Ordinary Riordan arrays 4 6 Production Matrices of Exponential Riordan arrays 4 7 Orthogonal polynomials and Riordan arrays 6 8 Introduction to the theory of Elliptic Functions 3 8 Ellipse arc length: 36 9 Elliptic functions derived from the A and Z generating functions of Riordan arrays 38 Jacobi Elliptic Functions and Riordan Arrays 4 Types of Jacobi Elliptic Functions and their properties 4 The Coefficient Arrays of the Jacobi Elliptic Functions 47 The Binomial transform in m 53 3 Jacobi Riordan arrays 57 4

8 3 Mixed subgroup of Jacobi Riordan arrays 58 3 Appell subgroup of Jacobi Riordan arrays The Associated subgroup of Jacobi Riordan arrays Derivative Subgroup of Jacobi Riordan arrays 76 4 Products of Jacobi Riordan arrays 87 4 [cn(z, m), sn(z, m)] 87 4 [cn(z, m), sn(z, m)] [ d dz sn(z, m), sn(z, m)] 88 [ 43 d dz sn(z, m), sn(z, m)] 90 [ 44 d dz sc(z, m), sc(z, m)] [ d dz sn(z, m), sn(z, m)] 9 [ 45 d dz sd(z, m), sd(z, m)] [ d dz sn(z, m), sn(z, m)] 9 [ 46 d dz sc (z, m), sc (z, m) ] [ d dz sn(z, m), sn(z, m)] 94 5 Jacobi Riordan arrays of complex variables with imaginary modulus k 94 5 The Riordan array from sn(iu, k ) 94 5 The Riordan array from cn(iu, k ) The Riordan array from dn(iu, k ) The Riordan array from cd(iu, k ) 98 6 Jacobi Riordan arrays from the A and Z generating functions 99 3 Weierstrass Elliptic Functions and Riordan Arrays 08 3 Introduction 08 3 Examples of Weierstrass Riordan arrays 3 Weierstrass Riordan arrays in terms of the Invariants {g, g 3 } 3 Weierstrass Riordan arrays in terms of the Modulus m 5 4 Dixonian Elliptic Functions and Riordan arrays 3 4 Introduction 3 4 Dixonian Riordan arrays 4 4 The Riordan array [ d dz sm(z), sm(z)] [ cm(z), sm(z) ] 5 4 The Riordan array [ d dz zcm(z), zcm(z)] 6 43 The Riordan array [, sm(z)] 7 44 The Riordan array [cm(z), z] 8 5 Embedded Sub-Matrices from a Riordan array 30 5 Introduction 30 5 The submatrices of Riordan [ arrays generated ] by Elliptic Functions3 5 The submatrix of dn(z,m) cn(z,m), z 3 5

9 [ ] 5 The submatrix of dn(z,m) cn(z,m), z 33 [ ] 53 The submatrix of dn(z,m) cn(z,m), z 34 [ 3 ] 54 The submatrix of dn(z,m) cn(z,m), z 35 [ 4 ] 55 The submatrix of dn(z,m) cn(z,m), z 35 [ 5 ] 56 The submatrix of dn(z,m) cn(z,m), z The submatrix of [ cn(z, m), sn(z, m) ] 36 [ sn(z,m) 58 The submatrix of, 37 +sn(z,m) ] 53 Embedded Riordan arrays from Riordan arrays generated by Elliptic Functions The embedded Riordan array of [cn(z, m), sn(z, m)] The embedded Riordan array of [dn(z, m), sn(z, m)] The embedded Riordan array of [cn(z, m), zcn(z, m)] The embedded Riordan array of [dn(z, m), zdn(z, m)] Riordan arrays derived from the r-shifted central triangles of a given Riordan [ array ] 40 z 54 cn(z, m), sn(z,m) 4 55 A one-parameter family of Riordan arrays derived from a given Riordan array 4 55 Examples using the inverse transformation T (s) 4 6 Riordan arrays and Solutions to Differential Equations 45 6 The Sturm-Liouville equation 45 6 Laguerre Polynomials 46 6 Hermite Polynomials The Reverse Bessel Polynomials Chebyshev Polynomials Morgan-Voyce Polynomials 59 6 Boubaker polynomials and the solutions of some Differential Equations 65 7 Elliptic Functions and the Travelling Wave Solutions to the KdV equation 69 7 Introduction 69 7 Elliptic functions as solutions to the KdV equation 70 7 Elliptic cn as a solution to the KdV equation 70 7 Elliptic Weierstrass as a solution to the KdV equation 74 6

10 73 Riordan arrays determined from Elliptic cn 76 [ 73 cn(ξ, m), cn(ξ, m) dξ ] 76 [ 73 cn(z, m), z ] 78 [ 733 cn(z, m), sn(z, m) ] The KdV Hierarchy 80 8 Riordan arrays inspired by the analytical solutions of nonlinear PDE of wave propagation in nonlinear low pass electrical transmission lines 8 8 Introduction ( ) ( ) 8 8 Generating functions from z(ξ) +z (ξ) & z (ξ) +z (ξ) 84 [ ] sn(ξ m) 83, sn(ξ m) + 85 [ ] 84 cn(ξ m) cn(ξ m) +, ξ 86 [ ] 85 sn(ξ m) sn(ξ m) +, ξ 87 [ ] 86, 87 ns(ξ m) ns(ξ m) + 87 Riordan arrays from cn(ξ m) sn(ξ m) and cn(ξ m) +sn(ξ m) 88 [ ] 87 cn(ξ m) +sn(ξ m), ξ 89 [ ] 88 z (ξ) +z (ξ), z(ξ) +z (ξ) Case : z(ξ) = sn(ξ m) Case : z(ξ) = ns(ξ, m) 96 9 Riordan arrays related to the FRLW Cosmological model 98 9 Introduction [ ( ) ] 98 k 9 Λ(+k cd η, k ) +k, η [a cd (η), η] 99 [ ( ) ] 93 Λ(+k dc η, k ) +k, η [a dc (η), η] 0 0 Riordan arrays and the analytical solution of the Quantum- Mechanical Oscillator Equation 03 0 Overview 03 0 The Harmonic Oscillator Quantum-Mechanical Oscillator Equation and Riordan arrays 05 Filter Design and Riordan arrays Introduction Systems Function of a Filter 7

11 The Order of a Filter 3 3 Types of Filters 4 4 Time and Group Delay 5 Elliptic Filter 5 3 Bessel Filter 8 3 Introduction [ 8 3 Bessel filter using t, ] t 9 [ 33 Bessel filter using 4t, ] 4t [ 34 Bessel filter using 6t, ] 6t 3 [ 35 Bessel filter using 3 3t, 3 ] 3t 4 [ 36 Bessel filter using 4 4t, 4 ] 4t 6 4 Passive Systems 7 4 Introduction 7 4 Riordan arrays from the Fibonacci and Morgan-Voyce polynomials 8 43 The Transfer function G(s) of Passive Sytems and Riordan arrays 3 5 Conclusion 3 Conclusions and Future Directions 34 Appendices 39 A Appendix-Symbolic Code 40 8

12 List of Figures A plot of Jacobi sn for m = /3 45 A plot of Jacobi cn for m = / A plot of Inverse Jacobi sn for m = / A plot of Inverse Jacobi cn for m = / Plot depicting a Weierstrass over a complex plane with g 3 = + i 0 3 The plot of Weierstrass elliptic function having {g, g 3 } = {, } 0 7 Plot depicting the KdV solution u(x, t) = sech (4t x) at t = Plot depicting the KdV solution u(x, t) = 4 3 sech (4t + x) at t = Plot depicting a Weierstrass solution for the KdV PDE 76 Bode plot of a third order lowpass elliptic filter model with cutoff frequency ω c = 7 Bode plot of a fourth order lowpass elliptic filter model with cutoff frequency ω c = 7 3 Bode Plot of a fifth order lowpass elliptic filter model with cutoff frequency ω c = 8 4 The plot shows the gain of low pass Bessel Filter of order n =,, 3, The plot shows the group delay of low pass Bessel filter of order n =,, 3, 4 6 The plot illustrates the effect of low pass Bessel filter of order 4 that filters out high frequency noise from a sinusoidal signal 7 The plot shows [ the gain of low pass Bessel Filter of order n =,, 3, 4 using 4t, ] 4t 9

13 8 The plot shows the[ group delay of low pass Bessel Filter of order n =,, 3, 4 using 4t, ] 4t 3 9 The plot shows [ the gain of low pass Bessel Filter of order n =,, 3, 4 using 6t, ] 6t 4 0The plot shows the[ group delay of low pass Bessel Filter of order n =,, 3, 4 using 6t, ] 6t 4 The plot shows [ the gain of low pass Bessel Filter of order n =,, 3, 4 using 3 3t, 3 ] 3t 5 The plot shows the[ group delay of low pass Bessel Filter of order n =,, 3, 4 using 3 3t, 3 ] 3t 6 3The plot shows [ the gain of low pass Bessel Filter of order n =,, 3, 4 using 4 4t, 4 ] 4t 7 4The plot shows the[ group delay of low pass Bessel Filter of order n =,, 3, 4 using 4 4t, 4 ] 4t 7 0

14 Chapter Introduction to Riordan Arrays and Elliptic Functions Background of the study In this chapter we shall discuss the theory of Riordan arrays and elliptic functions which are two of the main areas of research in this work The study of Riordan arrays uses extensively the interchangeable notions of sequences, formal power series, generating functions and in some cases the Lagrange inversion theorem These concepts are briefly described below A sequence is a mapping from the set N 0 of natural numbers into some other set of numbers such as the set of Real numbers R, Natural numbers N 0, Rational numbers Q, Complex numbers C That is for a mapping f the following holds f : N 0 R, f : N 0 N 0, f : N 0 Q, f : N 0 C In this case the sequences can be denoted as (f k ) k N0 In the case of a double sequence the mapping is such that f : N 0 N 0 Z

15 or mapped into any other well known numeric set of numbers There are two ways in which the mapping is denoted, which are {f n,k n, k N 0 } or (f n,k ) n,k N0 The most obvious application of a double sequence is in the display of an infinite array of numbers usually represented in matrix form The arrangement of the elements of the array is such that the first row can be generally seen as representing the sequence of numbers (f 0,0, f 0,, f 0,, f 0,3, ) with the second row given by (f,0, f,, f,, f,3, ) with the generalized form of the sequences from subsequent rows developed using a similar pattern The OEIS (Online Encyclopaedia of Integer Sequences)[00] provides a useful searchable database for researchers working in the area of enumerative combinatorics to identify useful named sequences which are the outputs of combinatorial processes or instances of combinatorial objects A formal power series (fps) f over a field of characteristic 0 in the indeterminate z is an expression of the form f(z) = f k z k = f 0 + f z + f z + f 3 z f n z n + k=0 where f n are the coefficients of the fps n N 0 A formal power series can also be regarded as an infinite degree polynomial The set of formal power series is denoted by F [[z]] or simply F The order of f(z), denoted ord(f(z)) is the smallest index r for which f r 0 The set of all formal power series having order exactly r is denoted F r The generating function[44] represents the closed form expression corresponding to a formal power series expansion encoding information about a sequence of numbers In enumerative combinatorics, the nature of the counting problem will determine if the generating function will be of either the ordinary or the exponential type The exponential type normally occurs when the order of the elements being counted is important The generating function is a more elegant way of working with sequences of varying levels of complexities in order to uncover useful patterns and for mathematical manipulation purposes Generating functions are sometimes called generating series as it can alternatively be viewed as a sequence of terms generating the sequence of term coefficients The Lagrange Inversion Formula can be applied in a variety of ways

16 [74, 88]: Determining the generating function of many combinatorial sequences In the extraction of the coefficients of a formal power series For the computation of combinatorial sums In the process of carrying out the inversion of combinatorial identities If w = w(t) is a formal power series satisfying the relation that w = tφ(w) with φ(0) 0 then the Lagrange inversion formula [63] states that [t n ]w(t) k = k n [tn k ]φ(t) n Setting f(t) = t/φ(t) = f(w(t)) = w/φ(w) = t, therefore w(t) can be considered the compositional inverse of f(t) The multivariate version of the LIF is given in [37] Some useful techniques for the extraction of coefficients in formal power series and the application of the Lagrange inversion formula have been applied to a variety of combinatorial problems [05] This thesis document will be divided into twelve chapters The first chapter will introduce the concept of Riordan arrays and will give a brief introduction to the notion of elliptic functions Chapters, 3, 4 will give the Riordan array representation of the three main types of elliptic functions seperately The Riordan arrays described in Chapter and 3 in particular, will be subdivided into various new classes of Riordan array subgroups parameterized by the elliptic modulus m In chapter 5 we will highlight interesting sub-matrices and new Riordan arrays that can be derived from some existing Riordan arrays based on their structural patterns In chapter 6 the relationship between Riordan arrays and the solutions to some useful systems of differential equations that have applications in mathematical physics and engineering will be highlighted as a preview to some other interesting applications in subsequent chapters Chapters 7, 8, 9 will identify Riordan arrays that are constructed based on the solutions arising from the applications of elliptic functions in mathematical physics Chapter 0 will focus on other applications of Riordan arrays which represent the solution of the quantum mechanical oscillator In chapter we will use Riordan arrays in the implementation of the Bessel filters and Elliptic filters in signal processing The last chapter will provide the main conclusions and the future direction 3

17 for further research based on the new results presented in this work There is also an appendix section with some relevant symbolic codes using Mathematica software from which the various investigations were carried out in support of this work A brief historical time-line on the development of Riordan arrays Riordan arrays were originally identified as a tool for solving combinatorial enumeration problems, particularly in lattice path theory [, 73, 98] The concept of Riordan arrays is most closely related to the Lagrange inversion theorem formulated by Lagrange [63] in 776 and to the umbral calculus developed in 970 s by S Roman [87] The points highlighted below give a summary of the key time-line in the development of the Riordan array theory In 978 Rogers [9] was inspired by L Shapiro s work which resulted in the discovery of the triangle of numbers called the Catalan triangle generated by its first column The Catalan triangle was the second type of array to be generated in a similar manner to the Pascal triangle which was previously the only known array with combinatorial properties This led to the formulation of a new method in determining the generalization of a family of triangular arrays called the Renewal arrays with arithmetic properties analogous to the Catalan and the Pascal triangle The result was the discovery of the A-sequence represented by a recursive formula (see [9]) that partially determined the entries of the new array known as the renewal array A follow-up to the work of Rogers on renewal arrays was the introduction in 99 of the Riordan Group [98] led by Louis Shapiro in collaboration with his colleagues Getu, Woan and Woodson from Howard Unversity, USA It was named in honour of John Riordan whose pioneering research work on combinatorics culminated in the publication of his well known book titled Combinatorial Identities [89] which is a useful resource on problems involving the inversion of combinatorial sums Shapiro[98] gives the initial definition of the properties of the Riordan group, its foremost theorem which is considered the fundamental theorem of Riordan arrays with the preliminary applications in evaluating combinatorial sums Fur- 4

18 thermore, the new theory of Riordan arrays focused on the columns and in particular algebraically describing them in terms of their associated generating functions rather than working with the umbral calculus Sprugnoli [0, 03, 7, 04] expanded the applications of Riordan arrays to the proofs of combinatorial identities, evaluation and inversion of combinatorial sums The introduction of the Z sequence characterization of Riordan arrays by Merlini et al (997) in [73] which completely solved the problem of recursively determining all the elements of a Riordan array The connection between production matrices and Riordan arrays by Emeric Deutsch [3] in 007 introduced the r and c sequence formulas analogous to the A and Z sequences to determine the recurrence coefficients and generating functions for the case of exponential Riordan arrays Previous research has found broader application of Riordan arrays to the following areas: Wireless communications for MIMO calculations [50] Computer science for algorithm analysis [75, 6] Molecular biology for RNA secondary structure enumeration [80] Error correcting codes in computer science [8] Chemistry [4] Queueing theory relating to birth-and-death processes [] Over the years there has been very active research with a multitude of publications in the area of Riordan arrays An evidence of this active research effort can be seen in the over 6300 results when the keyword Riordan arrays is entered into the Google search engine Sprugnoli in 008 [06] put together a bibliography collection of over 60 published research papers in the area This number has however increased significantly as can be seen from current updates available in online academic search engines showing that more research papers have been published on the subject Key researchers who have contributed enormously to the domain of Riordan arrays are: L Shapiro [98], R Sprugnoli [07], D Merlini [77], P Barry [4], A Luzón [70], M Morón [79], T-X He [49], GS Cheon [3] and so forth Their research contributions can be found on their 5

19 individual profiles available on academic web sites such as ResearchGate and Google Scholar from where their featured publications on Riordan arrays can be retrieved 3 Definitions and Examples of Riordan Arrays This section will provide the definitions to the three types of Riordan arrays together with some of their examples 3 Definitions of Riordan arrays The original forms of Riordan arrays comprise of the Ordinary and the Exponential Riordan array and these two types are the most often used A third type of Riordan array known as the Generalized Riordan array was introduced later Definition : An Ordinary Riordan array can be described as an infinite lower triangular array (d n,k ) 0 k n defined by a pair of formal power series represented by their analytic generating functions d(t) = d n t n and h(t) = h n t n such that the entries of the array columns are evaluated by the formula n=0 d n,k = [t n ]d(t)h(t) k where k is the column number It is denoted (d(t), h(t)) n=0 If all the conditions d(0) 0, h(0) = 0 and h (0) 0 are satisfied together then it is considered a proper Riordan array which shows that it is invertible In the case of the exponential Riordan arrays the definition is a different variation of the ordinary Riordan arrays given in terms of ordinary generating functions which are substituted in terms of exponential generating function Definition An Exponential Riordan array is an infinite lower triangular array consisting of a pair of formal power series represented by their exponential generating functions given by d(t) = + d n (t n /n!) h(t) = h n (t n /n!) n= n= 6

20 with the generic element associated with the coefficients of column k evaluated by It is denoted [d(t), h(t)] d n,k = n! k! [tn ]d(t)h(t) k From the above definition it is considered a proper exponential Riordan array otherwise it is considered a non-proper exponential Riordan array if h(0) 0 The importance of proper Riordan arrays will later be seen as a key condition in the sequence characterization of Riordan arrays which will be discussed in subsequent sections of this chapter Most of the applications of this thesis will be based on exponential Riordan arrays Alternatively, in terms of bivariate generating function an ordinary Riordan arrays [0] is given by d(t, x) = d(t)h(t) k x k d(t) = xh(t) () k=0 On the other hand the bivariate generating function of exponential Riordan arrays is given by d e (t, x) = k! d(t)h(t) k x k = d(t)e xh(t) () k=0 The case for x = in () and () results in the explicit formula for the row sum of a Riordan array Definition The definition of Riordan arrays associated with Laurent series is given in [47] as a Riordan matrix having entries given by d n,k = [t n ] d(t ) and denoted by (/d(t ), /h(t )) ( ) k h(t ) The most recently introduced type of Riordan array is the Generalized Riordan array which is a development of the ordinary and exponential Riordan arrays 7

21 Definition A Generalized Riordan array [5] consists of a Riordan array given by the pair (d(t), h(t)) expressed in relation to the non-zero sequence of elements c n with initial conditions c 0 = where d(t) = d k t k /c k h(t) = h k t k /c k k=0 such that its generic element from which its columns are generated can be expressed as c k k=0 [ ] t n d n,k = d(t)h(t) k A special feature arising from the definition of generalized Riordan arrays is that it is a more flexible concept to work with as it can be converted to the ordinary Riordan arrays if c n = and to exponential Riordan arrays if c n = n! 3 Some Basic Examples of Riordan arrays All Riordan arrays are a generalization of the Pascal triangle that is equivalent to the binomial matrix ( n k) which has all its leading diagonal elements equal to The most basic and important example of a Riordan array which gives a good entry point into investigating the various theories, methods and techniques arising from the Riordan array concept is the Pascal triangle The Pascal triangle consists of binomial coefficients which can be described in terms of its ordinary generating functions defined by ( ) t, t t with its general element given by ( n k) The generalized form of the Pascal triangle is given by ( ) ( ) m mt, t = mt t, t, m 0 t having general term ( n k) m n k Its equivalent exponential Riordan array is defined by [e t, t] The exponential type of Pascal triangle can be generalized as ([e t, t]) m = [e mt, t] such that m = encodes the binomial matrix entries We can apply the column definition of a Riordan array and the properties of binomial coefficients [44, 05] 8

22 to determine the general formula a n,k of the binomial matrix as illustrated below ( ) k a n,k = [t n t ] = [ t n k] t t ( t) k+ = [ t n k] ( ) (k + ) ( ) j t j j j=0 = [ t n k] ( ) k + + j t j j j=0 = [ t n k] ( ) k + j t j j j=0 ( ) k + n k = = n k ) = ( n n k ( ) n k This shows that a Riordan array is a generalization of the Pascal triangle Other well known examples of Riordan arrays that have received much attention are the Catalan arrays [9, 76] and ) the Fibonacci arrays [40] matrix is C = which has general term ( 4t t, 4t c n,k = k + ( ) n k n + n k The Catalan forming the Riordan matrix C = ( ) An example of a generalised Riordan array is ( + t ) λ0, t +t with respect ( ) to the sequence c n = ( λ n ) and t +t, t +t Several other examples of the generalised Riordan arrays are listed in [5] 9

23 4 The Group Structure of Riordan Matrices The main operation required to determine that a set of Riordan matrices forms a group is multiplication ( ) The multiplication rule for two Riordan arrays (g, f) and (h, l) is defined as (g, f) (h, l) = (g (h f), l f) (3) In Shapiro [98] the key criteria for (R, ) to be considered a group under the multiplication operation is stated based on the following conditions listed below The existence of (, t) as the identity element The inverse element is given by (d(t), h(t)) = ( ) d( h(t)), h(t) (4) We note that for a power series h(t) = n=0 h nt n with h(0) = 0, we define the reversion or the compositional inverse of h to be the power series h(t) such that h( h(t)) = h(h(t)) = t It can sometimes be denoted simply as h or Rev h 3 The set is closed under the multiplication of matrices 4 It is associative since multiplication of matrices is associative The next example shows the process of Riordan array multiplication and inverse computation ( Example Consider the Pascal triangle given by P = P P = = = t, ( ) ( ) t, t t t, t t ( ) t t t, t t t t ( ) t, t t ) t t 0

24 The compositional inverse of t P = = in P is t ( t +t ( + t, t + t t+ Therefore we have, ) t, + t ) The existence of the inverse element of Riordan arrays therefore makes them suitable to perform operations involving combinatorial sum inversion 4 The Sub-groups of the Riordan Group Some important sub-groups of the Riordan group have been presented in [99] The various forms of the subgroups arising from the Riordan array (d(t), h(t)) are listed as follows: The Appell subgroup has the form (d(t), t) The Lagrange subgroup has the form (, h(t)) The Lagrange subgroup is also known as the Associated subgroup The Renewal subgroup which is also referrred to as the Bell subgroup has the form (d(t), td(t)) The Hitting time( subgroup) was introduced by Peart and Woan in [84] and has the form th (t) h(t), h(t) where h (t) denotes the first derivative of h(t) The Checkerboard subgroup of the Riordan array (d(t), h(t)) represents a Riordan array such that d(t) is an even generating function and h(t) is an odd generating function The Derivative subgroup has the form (h (t), h(t)) The product of an Appell and a Lagrange Riordan array is such that (d(t), t) (, h(t)) = (d(t), h(t))

25 5 Sequence Characterization of Riordan arrays Three key theorems have underpinned much of the study within the concept of Riordan arrays They are the Fundamental Theorem of Riordan arrays together with the theorems on the A and Z sequences which characterizes the formation of a Riordan matrix The A and Z sequence characterization of a Riordan array (d(t), h(t)) involves determining both d(t) and h(t) respectively that define any such array The theorems arising from the sequence characterization of Riordan arrays present an alternative definition of a Riordan array in terms of the recursive formation of its elements These theorems are summarized below 5 The Fundamental Theorem of Riordan Arrays The Fundamental Theorem of Riordan arrays is an important gateway in proving many combinatorial identities and in solving problems related to combinatorial sums The Fundamental Theorem of Riordan arrays was put forward in the initial paper which introduced the concept of the Riordan group [98] The Fundamental Theorem of Riordan Arrays (FTRA) is formulated as follows: Theorem 5 [9] Suppose (d(t), h(t)) is a Riordan array Let A(t) = a k t k and B(t) = b k t k with A and B representing column vectors such that A = k=0 (a 0, a, a, ) T and B = (b 0, b, b, ) T Then (d, h) A = B, if and only if B(t) = d(t)a(h(t)) k=0 5 The A-Sequence The A-sequence introduced by Rogers (978) [9] characterizes the column elements after the first column Theorem 5 An infinite lower triangular array D = (d n,k ) n,k N0 is a Riordan array if and only if a sequence A = {a n } n N0 exists such that for every nk N 0 it is true: n k d n+,k+ = a i d n,k+i i=0

26 Even more, if D = (d(t), h(t)), then the generating function of the A sequence is such that A(t) = i=0 a it i satisfies the equation h(t) = ta(h(t)) A(t) = t h(t) 53 The Z-Sequence The Z sequence was introduced by Merlini et al and defined in the paper [73] as a follow-up study to completing the sequence characterization of Riordan arrays which started with the formulation of the A-sequence by Rogers in [9] The Z-sequence characterises the elements of the first column of a proper Riordan arrays as follows: Theorem 53 Let (d(t), h(t)) = (d n,k ) nk 0 Then a unique sequence Z = (z 0, z, z, z 3, ) can be determined such that every element in column 0 excluding the element in the first row can be expressed as a linear combination of all the elements in the preceding row with the coefficients identified as the elements of the sequence Z satisfying the relation n d n+,0 = z i d n,i (n N 0 ) i=0 The generating function of the Z sequence satisfies the equation d(t) = d 0 t(z(h(t)) ( = Z(t) = d ) 0 h(t) d( h(t)) 6 Production Matrices and Matrix Characterisation of Riordan Arrays This section presents an overview of the concepts of production matrices and the matrix characterization of Riordan arrays which will often be applied in subsequent chapters Production matrices were first introduced in the paper [30] Production matrices are simply described as infinite matrices that can be deduced from succession rules and which allows algebraic operations to be performed [7] The explicit formula for computing the production matrix P 3

27 associated to an infinite lower triangular matrix D = (d n,k ) is given by P = D D (5) where D represents the matrix corresponding to the top row of D deleted so that D = D n+,k The matrix characterization of Riordan arrays which results from the production matrix P provides an alternative view of the sequence characterization of Riordan arrays discussed in section (5) 6 Production Matrices of Ordinary Riordan arrays The production matrix P corresponding to an ordinary Riordan has the structure: z 0 a z a a z a a a P = = (Z(t), A(t), ta(t), t z 3 a 3 a a a 0 0 A(t), t 3 A(t), ) z 4 a 4 a 3 a a a 0 where (z 0, z, z, z 3, ) and (a 0, a, a, a 3, ) are the Z and A sequences corresponding to the generating functions Z(t) and A(t) respectively The algebraic structures of some subgroups of Riordan arrays are determined using their P matrix characterization [48] 6 Production Matrices of Exponential Riordan arrays The relationship between production matrices and Riordan arrays with the formulation of the r and c sequence characterisation of exponential Riordan arrays was established by Deutsch [3] These results are described in the following statements from [0] on the production matrices of exponential Riordan arrays Proposition 6 Let A = (a n,k ) n,k 0 = [g(x), f(x)] be an exponential Riordan array and let c(y) = c 0 + c y + c y +, r(y) = r 0 + r y + r y + (6) 4

28 be two formal power series such that Then r(f(x)) = f (x) (7) c(f(x)) = g (x) g(x) (8) (i) a n+,0 = i i!c i a n,i (9) (ii) a n+,k = r 0 a n,k + k! or, defining c = 0, i!(c i k + kr i k+ )a n,i (0) i k a n+,k = i!(c i k + kr i k+ )a n,i () k! i k Conversely, starting from the sequences defined by (6), the infinite array (a n,k ) n,k 0 defined by () is an exponential Riordan array A consequence of this proposition is that P = (p i,j ) i,j 0 where P = p i,j = i! j! (c i j + jr i j+ ) (c = 0) c 0 r !c!! (c 0 + r ) r !c!! (c + r )!! (c 0 + r ) r !c 3 3!! (c + r 3 ) 3!! (c + r ) 3! 3! (c 0 + 3r ) r 0 0 Furthermore, the bivariate exponential generating function of the matrix P is given by φ P (t, z) = n,k p n,k t k zn n! φ P (t, z) = e tz (c(z) + tr(z)) 5

29 Note in particular that we have r(z) = f ( f(z)) and c(z) = g ( f(z)) g( f(z)) 7 Orthogonal polynomials and Riordan arrays The modern theory of orthogonal polynomials is based on the research work by Szegö [0] The importance of orthogonal polynomials (OPs) are most often associated to the solutions of mathematical and physical problems These polynomials have been identified in areas of study such as wave mechanics, heat conduction, electromagnetic theory, quantum mechanics, electronic filter design and mathematical statistics [83] Definition An orthogonal polynomial sequence(ops) (p n (x)) n 0 refers to a sequence of polynomials p n (x), n 0 of degree n, with real coefficients such that any pair in the sequence are mutually and continuously orthogonal on an interval [x 0, x ] with respect to a weight function ω : [x 0, x ] R: x x 0 p n (x)p m (x)ω(x)dx = δ nm hn h m where δ nm = { if n = m 0 if n m and x x 0 p n(x)ω(x)dx = h n We assume that ω is strictly positive and continuous on the interval (x 0, x ) Additionally, the weight function ω satisfies the first order differential equations with polynomial coefficients Some examples of classical orthogonal polynomials are the Hermite, Laguerre, Legendre, Jacobi, Bessel, Chebyshev [35] Some of these polynomials will be treated in subsequent chapters These systems of orthogonal polynomials have the following common properties below [8] The weight function ω(x) on the interval of orthogonality (a, b) satisfies 6

30 the Pearson differential equation ω (x) ω(x) = p 0 + p x q 0 + q x + q x A(x), x (a, b) B(x) where the following conditions hold at the end points of the interval of orthogonality: lim ω(x)b(x) = lim ω(x)b(x) = 0 x α+0 x b 0 The polynomial y = P n (x) of order n satisfies the differential equation B(x)y + [A(x) + B (x)]y n[p + (n + )q ]y = 0 3 The Rodrigues formula holds: c n d n P n (x) = ω(x) dx n [ω(x)bn (x)] A key characteristic of sequences from a family of orthogonal polynomial sequences is that they satisfy a so-called three term recurrence [83] Theorem 7 [9] A sequence of orthogonal polynomials {p n (x)} n=0 satisfies p n+ (x) = (γ n x α n )p n (x) β n p n (x) for coefficients γ n, α n and β n that depend on n but not xwe note that if p (x) = k j x j + k jx j + j = 0,, then γ n = k ( n+ k ) ( ), α n = γ n+ n k n kn h n, β n = γ n k n k n+ k n k n h n Since the degree of p n (x) is n, the coefficient array of the polynomials is a lower triangular (infinite) matrix In the case of monic orthogonal polynomials the leading coefficient of the polynomial k n = for all n The moments associated to the orthogonal polynomial sequence are the numbers µ n = x x 0 x n w(x)dx 7

31 p n (x), α n,β n and w(x) can be determined from known moments If n denotes the Hankel determinant µ i+j n i,j 0 and n,x denotes the same determinant, but with the last row equal to, x, x, then p n (x) = n,x n () ( ) u u k More generally, let H be the determinant of Hankel type with v v k (i, j)-th term µ ui+v j Let n = H ( 0 n 0 n ), n = H n ( 0 n n 0 n n + ) Thus, α n = n n n n, β n = n n n In summary for a family of polynomials {p n (x)} n 0 to be formally orthogonal, then there exists a linear functional L on polynomials satisfying the following conditions: p n (x) is a polynomial of degree n, L(p n (x)p m (x)) = 0 for m n, 3 L(p n(x)) 0 4 The sequence of numbers µ n = L(x n ) is called the sequence of moments of the family of orthogonal polynomials defined by L 5 There exists an OPS wrt L if and only if n 0, n N 0 In particular, for the given moment sequence (µ n ) n=0 µ 0 µ µ n µ µ µ n+ n = det[µ j+k ] n j,k=0 = µ n µ n+ µ n 8

32 The following well-known results listed below specify the links between orthogonal polynomials, three term recurrences, the recurrence coefficients and the gf of the moment sequence of the orthogonal polynomials Theorem 7 [4, 7], Theorem 50)(Favard s Theorem) Let (p n (x)) n 0 be a sequence of monic polynomials, the polynomial p n (x) having degree n = 0,, Then the sequence (p n (x)) is (formally) orthogonal if and only if there exist sequences (α n ) n 0 and (β n ) n with β n 0 for all n, such that the three-term recurrence p n+ = (x α n )p n (x) β n p n (x), for n, holds, with initial conditions p 0 (x) = and p (x) = x α 0 Theorem 73 [4], Theorem 5) Let (p n (x)) n 0 be a sequence of monic polynomials, which is orthogonal with respect to some functional L Let p n+ = (x α n )p n (x) β n p n (x), for n, be the corresponding three-term recurrence which is guaranteed by Favard s theorem Then the generating function g(x) = µ k x k k=0 for the moments µ k = L(x k ) satisfies g(x) = α 0 x α x Given a family of monic orthogonal polynomials µ 0 β x β x β 3 x α x α 3 x p n+ (x) = (x α n )p n (x) β n p n (x), p 0 (x) =, p (x) = x α 0, we can write n p n (x) = a n,k x k k=0 9

33 Then we have n+ n n a n+,k x k = (x α n ) a n,k x k β n a n,k x k from which we deduce k=0 k=0 k=0 a n+,0 = α n a n,0 β n a n,0 (3) and a n+,k = a n,k α n a n,k β n a n,k (4) We note that if α n and β n are constant, equal to α and β, respectively, then the sequence (, α, β, 0, 0, ) forms an A-sequence for the coefficient array Proposition 74 [85] If L = (g(x), f(x)) is a Riordan array and P = S L is tridiagonal, then necessarily where and vice-versa a b a b a 0 0 P = S L = 0 0 b a b a b a x f(x) = Rev + ax + bx and g(x) = a x b xf, This leads to the important corollary Corollary 75 [] If L = (g(x), f(x)) is a Riordan array and P = S L is 30

34 tridiagonal, with P = S L = a b a b a b a b a b a then L is the coefficient array of the family of orthogonal polynomials p n (x) where p 0 (x) =, p (x) = x a, and p n+ (x) = (x a n )p n (x) b n p n (x), n, where b n is the sequence, b, b, b, b, Theorem 76 [] A Riordan array L = (g(x), f(x)) is the inverse of the coefficient array of a family of orthogonal polynomials if and only if its production matrix P = S L is tri-diagonal Proposition 77 Let L = (g(x), f(x)) be a Riordan array with tri-diagonal production matrix S L Then [x n ]g(x) = L(x n ), where L is the linear functional that defines the associated family of orthogonal polynomials Based on (77) the moment sequence is given by the first column of the Riordan array L whenever the conditions established for the orthogonality of a Riordan array is satisfied 8 Introduction to the theory of Elliptic Functions In the mathematical area of complex analysis, an elliptic function is considered to be a function of a complex variable which is meromorphic on an open set and 3

35 is doubly periodic [8, 58] On the other hand, an elliptic integral from which the development of elliptic functions originates, represents an integral of the form R(x, p(x))dx where R(x, w) is a rational function in two variables and p(x) is a polynomial of degree 3 or 4 having no repeated roots The development of the theory of elliptic functions is historically linked to earlier theoretical studies of elliptic integrals beginning from the 7 th century as outlined in [39] During this period many interesting integrals arose during the process of solving mechanical problems One of the most notable of such problems was when Wallis in 655 began studying the arc length of an ellipse by considering the lengths of various cycloids and then relating them to the arc length of an ellipse This culminated in the publication of an infinite series expansion of the arc length of an ellipse This result was followed by the work of the mathematician Jacob Bernoulli ( ) who in 694 extended the results on the theory of elliptic integrals The result of J Bernoulli derived from his experiment on compressing an elastic rod at its ends, showed that the resulting curve satisfied the equation [39] ds dt = t 4 This was followed by the introduction of the lemniscate curve given by (x + y ) = (x y ) The arc length of the lemniscate curve was subsequently determined by computing an elliptic integral referred to as the lemniscate integral given by x 0 dt t 4 The above integral can be expressed in terms of arcsine x given by sin (x) = x 0 dt t Another important observation by Jacob Bernoulli in 694 found that the inte- 3

36 gral t dt t 4 could not be expressed in terms of existing transcendental functions This nature of problem served later as one of the motivations for further research on such integrals The origins of the main theory of elliptic functions can be traced back to the 9th century work on integral calculus by the famous mathematician Niels H Abel (80 89) [5] His most remarkable achievement in this area was implementing the main technique of inverting elliptic integrals which led to elliptic functions Abel using the main results of Euler, Lagrange and Legendre on elliptic integrals defines his elliptic function ϕα = x published in his memoir titled Recherches Sur Les Fonctions Elliptiques(88)[] by the relation α = x o dt ( c t )( + e t ) (5) where c and e are real numbers In addition, the definition (5) is equivalent to the differential equation ϕ α = ( c ϕ α)( + e ϕ α) Abel extends the definition of ϕα = x to the entire complex domain using the addition theorem based on Euler addition theorem for elliptic integrals The main motivation to investigate the inverse of elliptic integrals was due to the three forms of elliptic integrals put forward by Legendre in his publication Traité des fonctions elliptiques et des intégrales eulériennes (Paris,85)[67] These forms are F (φ, k) = φ 0 E(φ, k) = φ 0 3 Π(φ, n, k) = φ dφ x k sin φ 0 k sin φdφ x 0 dφ 0 (+n sin φ) k sin φ x dx 0 (+nx ) ( x )( k x ) dx (if x = sin φ) ( x )( k x ) (if x = sin φ) k x x dx (if x = sin φ) which is equivalent to These integrals are called elliptic integrals of the first, second and third kind respectively These integrals were the outcome of the effort by Legendre to 33

37 compute the arc length of an ellipse NH Abel (80 89) contributed by inverting the Legendre elliptic integral of the first kind This led to the introduction of the possibility of working with complex variables rather than the restriction to only real variables The inverse function derived from inverting the elliptic integral is considered the simplest elliptic function During the same period when Abel put forward his work on the inversion of elliptic integrals another mathematician Carl G Jacobi(804 85) also worked in the same area In 89 Jacobi introduced the Jacobi elliptic functions denoted as sn u, cn u, dn u A key characteristic of these functions is that they satisfy the equation describing quartic elliptic curves given by (y ) = ( x )( k x ) Jacobi s work on elliptic functions primarily focused on the inverse function as presented in the sequence below: φ = am u (am u is the Jacobi amplitude function) of the integral of the first type u = φ 0 dφ k sin φ This leads to the 3 single valued functions derived from the inverse function am u which is multivalued, these functions are sin φ = sin am u cos φ = cos am u φ = am u The standard notation of Jacobi elliptic functions are denoted from the results above as follows sin am u sn u cos am u cn u am u dn u The Jacobi elliptic functions may be explicitly defined by first defining sn u 34

38 sn(x, k) such that sn(x, k) = Rev ( x 0 ) dt ( t )( k t ) Jacobi s work on elliptic functions led to his well known publication titled Fundamenta nova theoriae functionum ellipticarum (89) in which he presented the four theta functions [55] The most remarkable achievement of Jacobi was establishing the relation between the elliptic theta functions that are closely related to the elliptic functions [45] The Jacobi theta functions are the elliptic analogs of the exponential functions which can be expressed in terms of the Jacobi elliptic functions Following on from the work of Jacobi on ellliptic functions, the mathematician Karl Weierstrass (85 897) also made significant contributions to the theory of elliptic functions His main contributions was introducing new functions which behave in a simpler way In particular, he replaced the 3 basic types of Jacobi elliptic functions sn u, cn u and dn u by the single function known as the Weierstrass which is also referred to as the (Weierstrass P ) function Similarly, he replaced the Jacobi theta functions by the sigma and the zeta functions The Weierstrass elliptic functions are known to parametrize cubics given in the Weierstrass form by (y ) = 4x 3 g x g 3 A third type of elliptic function known as the Dixon s elliptic functions was introduced by the British mathematician Alfred Cardew Dixon ( ) His main work on elliptic functions is collected in his publication The elementary properties of the elliptic functions, with examples (894) [33] There are few available research publications on Dixon elliptic functions compared to the Jacobi and Weierstrass elliptic functions that have received considerable research focus to date The most recent publication on Dixon elliptic functions is the work by Langer & Singer (03)[65] in which the arc length of a sextic curve referred to as the trefoil is expressed in terms of Dixon elliptic functions A prior publication by Conrad and Flajolet(006) [6] extensively covers the combinatorial aspects and certain pattern permutations arising from a study of these functions 35

39 The first six terms of the coefficients of the Taylor series expansion of sn(x, k) = Rev ( x 0 ) dt ( t )( k t ) are { 0,, 0, k, 0, k 4 + 4k +, 0 } and for are arcsn(x, k) = x 0 dt ( t )( k t ) { 0,, 0, k +, 0, 3 ( 3k 4 + k + 3 ), 0 } We note that we can find the power series coefficients of sn(x, k) in the first column of the inverse Riordan array [ arcsn(x, k), arcsn(x, k)] x More recent work over the last century has focused on aspects of the Taylor series coefficients of elliptic functions such as getting recurrence formulas and combinatorial interpretations with enumerative properties [4, 94, 95, 6, 6] Elliptic functions have also been studied in relation to their connection to orthogonal polynomials [53, 54] Using the aid of concrete examples we shall relate elliptic functions to exponential Riordan arrays in the next three chapters More details pertaining to each of these functions will be elaborated in the subsequent chapters For the purpose of this thesis the Riordan arrays generated from the Jacobi elliptic functions, Dixonian elliptic functions and Weierstrass elliptic functions respectively will simply be referred to as Jacobi Riordan arrays, Dixonian Riordan arrays and Weierstrass Riordan arrays respectively 8 Ellipse arc length: We begin with x a + y b = 36

40 Thus We also get Thus, x a = x y = y = b xdx a + ydy b = 0 ( x a ) dx + dy = dx + x dx y b 4 a 4 = dx + x dx b4 a 4 b ( x a ) = dx + x dx b a a x ( ) = dx + x b a a x ( (a = dx x ) + x b a a x ( ( x = dx a ) + x b a 4 x a ( ) x = dx a b a x a ) ) Thus, s = = = = a x 0 x 0 x 0 x a dx + dy x a ( b x a x a ( a b a ) a ) x a k t t dt where k = a b a and we have set t = x a Note that 0 < k < 0 dx dx 37

41 9 Elliptic functions derived from the A and Z generating functions of Riordan arrays In this section the relationship betweeen elliptic functions and the A and Z generating functions of Riordan arrays is established This relationship serves as the key motivation for further research on elliptic functions and Riordan arrays that has previously not been investigated We know that a Riordan array can be defined in terms of its generating functions But an alternative definition of Riordan arrays can be described in terms of its A and Z sequences and their corresponding generating functions [3] That is for a given pair of generating functions g(x) F 0 and f(x) F where x is an indeterminate, we can define the Riordan array symbolically as [g(x), f(x)] In terms of the A and Z generating functions associated to the Riordan matrix corresponding to [g(x), f(x)], we can symbolically define the Riordan matrix as [g 0, A; Z] We let M = [g, f] denote an exponential Riordan array The production matrix of M is the matrix M M, where M is the matrix M with the top row removed The array M determines and is determined by its production matrix The production matrix has bivariate generating function e xy (Z M (x) + ya M (x)), where we have and A M (x) = f ( f(x)) Z M (x) = g ( f(x)) g( f(x) For the inverse exponential Riordan array M = [g, f], we have A M = f (x), and Z M = g (x) f (x) g(x) We can express the array [g, f] in terms of A = A M and Z = Z M as follows 38

42 [ x [g(x), f(x)] = e 0 Z(Rev( t 0 Alternatively, we can write [g(x), f(x)] = [ e Rev ( x0 0 dt A(t)) ( x A(t)))dt dt, Rev 0 ( Z(t) x A(t) dt, Rev 0 )] dt A(t) ) ] dt A(t) Now we recall that an integral is called an elliptic integral if it is of the form R(x, P (x)) dx, where P (x) is a polynomial in x of degree three or four and R is a rational function of its arguments Thus if A(t) = R(t, P (t)), then the above exponential Riordan array can be said to be defined by an elliptic integral Elliptic functions are defined as the inverses of elliptic integrals Thus the expression ( x ) dt Rev 0 A(t) in the defining relation for [g, f] is an elliptic function when A(t) = R(t, P (t)) In this case we will have where A(t) = will also be a rational function R(t, P (t)) = R(t, P (t)), R = R It is therefore natural to call an exponential Riordan array M = [g, f] an elliptic Riordan array if A M (t) = R(t, P (t)), where R is a rational function and P (t) is a polynomial of degree three or four Lagrange showed that any elliptic integral can be written in terms of the 39

43 following three fundamental or normal elliptic integrals F (x, k) = Π(x, α, k) = x 0 E(x, k) = x 0 dt ( t )( k t ), x 0 k t t dt, dt ( α t) ( t )( k t ) These integrals are called elliptic integrals of the first, second and third kind, respectively As an example of elliptic functions, the Jacobi elliptic functions may be defined by first defining sn(x, k) as follows, sn(x, k) = Rev ( x 0 ) dt, ( t )( k t ) involving the elliptic integral of the first kind, and then we define cn(x, k) = sn(x, k) and dn(x, k) = k sn(x, k) Thus if A(t) = ( t )( k t ), ie A(t) = ( t )( k t ), then we obtain an exponential Riordan array with f(x) = sn(x, k) 40

44 Chapter Jacobi Elliptic Functions and Riordan Arrays In this chapter we shall begin by giving an introduction of the Jacobi elliptic functions This will be followed by various examples of Riordan arrays generated from these functions Types of Jacobi Elliptic Functions and their properties The initial point to understanding elliptic functions is from the perspective of the two basic trigonometric functions of the sine and cosine The elliptic function is considered a generalization of these functions since its most basic form reduces to a trigonometric function The basic properties of the various types of Jacobi elliptic functions can be traced back to the trigonometric sine and cosine functions The parity property of these functions will later become useful in the construction of a new class of Riordan arrays which is referred here as Jacobi Riordan arrays The Riordan arrays generated from these functions can be viewed to fall under the Double Riordan group [9] and in particular the checkerboard subgroup of Riordan arrays The Jacobi elliptic functions were originally derived from the elliptic integral of the first kind [58] We can define the Jacobi elliptic function sn using a reversion technique, beginning with what will be its reversion such that 4

45 sn(z, k) = Rev which can be rewitten as arcsn(z, k) = ( z z 0 0 ) dt, ( t )( k t ) ( k t )( t ) dt An alternative method for defining sn(z, k) is to start with the function F (φ, k) = φ 0 k sin(θ) dθ arcsn(z, m) u(ϕ, m) = sin ϕ 0 dz ( z )( mz ) () where m = k The parameter k where 0 k < is called the modulus of the elliptic integral and ϕ is called the amplitude The complementary modulus is k = k The Legendre form of the elliptic integral which is an alternative definition of elliptic sn(z, m) is given by arcsn(z, m) u(ϕ, m) = ϕ 0 dθ m sin θ () Equation () follows from a change of variables which can be determined from () using the substitution z = sin θ such that dz = cos zdθ = z dθ We then revert the function F to get the amplitude function am(u, k) = F (u, k) = φ Finally we define sn(u, k) = sin(am(u, k)) = sin(φ) The two forms of the elliptic integral u in () & () are also known as the incomplete elliptic integrals On the other hand the complete elliptic integrals are given by ( π ) K(m) := u, m dz = 0 ( z )( mz ) 4

46 ( π ) π u, m = 0 dθ m sin θ with its complete complementary elliptic integral defined as K (m) := u ( π, m ) = 0 dz ( z )( ( m)z ) u ( π, m ) = π 0 dθ ( m) sin θ where m + m = & m = (k ) & m & K < z < K The hypergeometric interpretation of the complete elliptic integral is defined by K(m) = π F (, ; ; m ) K (m) = π F (, ; ; m ) The Jacobi elliptic functions are derived by inverting the elliptic integral () The three basic Jacobi elliptic functions from which other forms are obtained are: the elliptic sine sn(u;m)= sin ϕ, the elliptic cosine cn(u;m)= cos ϕ and the difference function dn(u;k)= dϕ du The basic properties of the elliptic Jacobi functions are analogous to those of the well known trigonometric functions The basic ellliptic functions satisfy the equation sn (u; m) + cn (u; m) = from the trigonometric relation sin ϕ + cos ϕ = Thus, sn(u; m) = cn (u; m) & dn(u;m) = dϕ du = msn (u; m) All three functions sn(u, k), cn(u, k) and dn(u, k) are doubly periodic The following holds true for Jacobi elliptic functions: 43

47 sn(u + 4K, k) = sn(u, k), where sn(k, k) = sn(u + L, k) = sn(u, k), where cs(l, k) = i cn(u + 4K, k) = cn(u, k) cn(u + 4L, k) = cn(u, k) dn(u + 4K, k) = dn(u, k) dn(u + 4L, k) = dn(u, k) dn (u, k) + k sn (u, k) = In addition, the three basic forms of the Jacobi elliptic function determine the other 9 forms of Jacobi elliptic functions such that the definitions are quotients of any of these three These twelve forms of Jacobi elliptic functions are: s c d n s sc sd sn c cs cd cn d ds dc dn n ns nc nd For example sd(u; m) = sn(u;m) dn(u;m) and similar results apply to each of the other forms shown on the table The basic properties of the Jacobi elliptic functions are: sn(0; k) = 0 sn(k; k) = cn(0; k) = cn(k; k) = 0 dn(0; k) = dn(k; k) = k In the limit we have, 44

48 lim sn(u, m) m 0 = sin(u) cn(u, m) = cos(u) lim m 0 lim m 0 lim m lim m lim m dn(u, m) = sn(u, m) = tanh(u) cn(u, m) = sech(u) dn(u, m) = sech(u) The Jacobi elliptic functions satisfy the differential system: for the initial conditions d sn(u, m) du = cn(u, m)dn(u, m) d cn(u, m) du = dn(u, m)sn(u, m) d dn(u, m) du = msn(u, m)cn(u, m) sn(0, m) = 0, cn(0, m) = dn(0, m) = The systems of differential equations satisfied by the Jacobi elliptic functions forms the basis of their applications to some dynamical systems [78] The plots corresponding to Jacobi sn, Jacobi cn, Inverse Jacobi sn, Inverse Jacobi cn respectively for m = /3: Figure : A plot of Jacobi sn for m = /3 45

49 Figure : A plot of Jacobi cn for m = / Figure 3: A plot of Inverse Jacobi sn for m = / Figure 4: A plot of Inverse Jacobi cn for m = /3 46

50 The Coefficient Arrays of the Jacobi Elliptic Functions By reviewing the Jacobi elliptic function arcsn(z, k) in section (), we find that [0, 5] arcsn(z, k) = = = = = = z 0 z 0 z 0 z ( k t )( t ) dt (k + )t + k t 4 dt 0 n=0 z 0 k + ( k n P n n=0 ( k + P n k ( k + P n k n=0 ( k + k dt u = kt u k + u ) u n dt ) k n t n dt )) z n+ n + by The first of the power series coefficients for arcsn(x, k) are therefore given { 0,, 0, m +, 0, 6 40 ( ) 3m + m + 3, 0, } ( ) 5m 3 + 3m + 3m + 5, 0, 35m4 + 0m 3 + 8m + 0m + 35, 0, 5 where we write m = k It is clear from the power series expansion above that arcsn(0, k) = 0 for all k, and hence we can revert this power series to obtain the reversion of arcsn(x, k), which is the Jacobi elliptic function sn(x, k) The first power series coefficients for sn(x, k), where we use m = k, are as follows { 0,, 0, 6 ( m ), 0, 0 ( ) m + 4m +, 0, m3 35m 35m 5040 }, 0, m4 + 8m m + 8m +, Thus we have sn(x, k) = x 3! ( + k )x 3 + 5! ( + 4k + k 4 )x 5 We see that sn(x, k) is an odd function We shall spend some time studying these coefficients Ignoring the zero 47

51 entries for the moment, we note that the denominator sequence begins, 6, 0, 5040, 36880, , , , , This is the sequence (n + )! We now look at the polynomial sequence in m given by the numerators of the power series coefficients of sn(x, k) We shall multiply this sequence by ( ) n to simplify it We obtain the sequence in m that begins { }, m +, m + 4m +, m m + 35m +, m 4 + 8m m + 8m +, The coefficient array for this sequence of polynomials begins We note immediately that this is a symmetric triangle, reminiscent of Pascal s triangle and of the Narayana triangle Narayana triangle is not an accident Its resemblance in form to the We recall that the Narayana triangle has a continued fraction bivariate generating function given by x xy x xy x y x y x xy x y It turns out that the above sn(x, k) inspired triangle has a similar type bivariate generating function 48

52 x xy 9x 9xy x y 5x 5xy 40x y 60x y 49x 49xy 403x y The coefficients in this fraction are based on the sequence,,, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, where the numbers are taken in groups of 4 ( 3 = etc) for the β coefficients, and two by two for the α coefficients (3 3 = 9 etc) This J-fraction [5] may also be written equivalently as follows: x 9 + xy x x 9xy x 9 x 5xy 60 5 x x 49xy, or 49

53 x xy x + 3 x 9xy 40 9 x x 5xy 60 5 x x 49xy The Hankel transform of the sequence, m +, m + 4m +, m m + 35m +, can be calculated to be [5] n+ h n = k n(n+) i=0 i! We note that there is a recurrence for the coefficients b(n, k) = n!a(n, k), where a(n, k) = [x n ]sn(x, k) which can be described as follows [9] 0 if n is even, if n =, b(n, k) = (k + ) if n = 3, k 4 + k + if n = 5, f(n) if n > 5, n odd where 50

54 (n 5)f(n) = (n )( + k )b(n, k) + + (( ) ( )) n + n b(i, k)b(n + i, k) 3 i i n i= ( ) n ( + k ) b(i, k)b(n i, k) i n i= The power series coefficients for cn(x, k) begin {, 0,, 0, m 6 + 4, 0, 70 and so we have } ( ) 6m 44m, 0, 64m3 + 9m + 408m cn(x, k) =! x + 4! ( + 4k )x 4 6! ( + 44k + 6k 4 )x 6 + We see that cn(x, k) is an even function The adjusted sequence of coefficients (multiplying by the denominators multiplying by ( ) n as appropriate) gives us the polynomial sequence,, +4m, 44m 6m, +408m+9m +64m 3, 3688m 30768m 5808m 3 56m 4, which has the coefficient array This array has the following continued fraction bivariate generating function 5

55 x 4xy 9x 6xy 5x The Hankel transform of the sequence,, +4m, 44m 6m, +408m+9m +64m 3, 3688m 30768m 5808m 3 56m 4, can be calculated to be [5] ( n h n (m) = m ) (n (k )!) k= The coefficients for the power series of dn(u, k) begin, 0, m, 0, 4 ( ) m + 4m, 0, 70 ( ) m 3 44m 6m, 0, m m 3 + 9m + 64m, 4030 and thus we have dn(x, k) =! k x + 4! (4k + k 4 )x 4 The function dn(x, k) is an even function From it we obtain the polynomial sequence, m, m + 4m, m 3 44m 6m, m m 3 + 9m + 64m, from which (dropping signs) we derive the coefficient array 5

56 This array has the continued fraction bivariate generating function given by or equivalently, xy 4x 9xy 6x 5xy, xy 4x 9xy 4x y 44x y 6x 5xy 400x y The Binomial transform in m The matrix [5]

57 is more correctly associated to the Jacobi elliptic function dc as it is the coefficient array of (n)![x n ]dc(x, m + ) Multiplying this matrix on the right by the binomial matrix B is the same as letting m m + We get B = This is the coefficient array associated to the Jacobi elliptic function dc(x, m) More precisely, it is the coefficient array for (n)![x n ]dc(x, m) This array has bi-variate generating function given by x(y + ) 4x 9x(y + ) 6x 5x(y + ) 54

58 In like manner, the array B = , is the array associated to (n)![x n ]dc(x, m ), with bi-variate generating function x(y + ) 4x 9x(y + ) 6x 5x(y + ) 55

59 In a similar manner, the array is given by n!( ) n [x n+ ]sn(x, m), with bivariate generating function x 9 + xy x x 9xy x 9 x 5xy x 5 x 49xy x Then the array B = 56

60 has bivariate generating function x + 3x x(y + ) x 9x(y + ) 40 9 x x 5x(y + ) 60 5 x 49x(y + ) + 7 5x It is the array for n!( ) n [x n+ ]sn(x, m + ) 3 Jacobi Riordan arrays In this section Riordan arrays are constructed based on the Jacobi elliptic functions in terms of the elliptic modulus m under their various subgroups A combinatorial interpretaion for each of these matrices corresponding to their trigonometric and hyperbolic generating functions will be noted using the OEIS online resource The possible combinatorial interpretations of the Taylor series coefficients of the Jacobi elliptic functions sn and cn have been investigated in [4, 94, 95, 9] Using the Taylor series coefficients of these functions we shall proceed with the construction of Riordan matrices with their corresponding production matrices to get an alternative view in the study of such functions 57

61 3 Mixed subgroup of Jacobi Riordan arrays 3 [cn(z, m), sn(z, m)] The coefficient matrix of [cn(z, m), sn(z, m)] begins A = 0 6 ( m 6 ) ( m 6 + ) 4 0 ( m 3 ) m + 44m ( m ) 0 0 6m 44m 0 6m + 4m ( m 3 ) which is equivalent to A = 0 m m + 0 (m + 5) m + 44m (m + ) 0 0 6m 44m 0 6m + 4m (4m + 7) 0 Remark: The row sums for m = 0 form the sequence (,, 0, 3, 8, 3, 56, ) which has egf cos(z)e sin(z) The row sums for m = form the sequence (,, 0, 4, 8, 3, 6, ) corresponds to A00965 with egf e tanh(z) cosh(z) The production matrix of A in terms of m : C = 0 m (m ) 0 3(m + ) ( m 6m + 5 ) 0 (3m + 5) 0 5 ( m + m 3 ) 0 5 ( m 4m + 3 ) 0 5(m + 3) 0 If m = 0,, then [cn(z, m), sn(z, m)] produces the Riordan arrays D = {[cos(z), sin(z)], [cn(z, ), sn(z, )], [sech(z), tanh(z)]} respectively The production matrices from C for m=-,0, associated to the Riordan arrays in D are as follows: E = , ,

62 Remark: The production matrix in E at m = is tridiagonal [ which indicates] that the inverse of [cn(z, ), sn(z, )] = [sech(z), tanh(z)] =, z tanh (z) is the coefficient array of a family of orthogonal polynomials The three term recurrence relation for the family of orthogonal polynomials is given by P n+ (z) = zp n (z) + n P n (z), n with P 0 (z) =, P (z) = z In particular, let Q n (z) = P n(iz) i n, (i = ) we get Q n+ (z) = zq n (z) n Q n (z), n The generating functions of the r and c sequences associated to the production matrices of D and C are listed below: r(z, m) = cn ( sn (z m) m ) dn ( sn (z m) m ) r(z, ) = z r(z, 0) = z c(z, m) = zcn ( sn (z m) ) ( m dn sn (z m) ) m c(z, ) = z c(z, 0) = z z z Alternatively, we can derive the r and c generating function of [cn(z, m), sn(z, m)] using the elliptic integrals defined in () as follows: c(z) A(z) = sn (sn(z, m)) & Z(z) = cn (sn(z, m)) cn(sn(z, m)) 59

63 For A(z) = sn (sn(z, m)) = sn ( z 0 ) ( mu )( u ) du = cn(sn(z, m))dn(sn(z, m)) = ( sn (sn(z, m)))( msn (sn(z, m))) = ( z )( mz ) For r(z) Z(z) = cn (sn(z, m)) cn(sn(z, m))) sn(sn(z, m))dn(sn(z, m)) = cn(sn(z, m)) = z mz z 3 [cd(z, m), sn(z, m)] The coefficient matrix of [cd(z, m), sn(z, m)] is given by m A = 0 (m ) m 6m + 0 (m 5) ( m m + ) m 3 07m + 47m 0 3m + 4m (m + 7) 0 Remark: The row sums of A for m = 0 form the sequence (,, 0, 3, 8, 3, 56, ) which has the egf cos(z)e sin(z) The row sums of A for m = form the sequence (,,,, 7, 3, 97, ) corresponds to A00373 with egf e tanh(z) 60

64 The production matrix of A in terms of m is given by m B = 0 m ( m ) m + 6m 5 0 (m + 5) 0 5 ( 3m 3 m + m 3 ) 0 5 ( m + m 3 ) 0 5(m + 3) 0 If m =, 0, then [cd(z, m), sn(z, m)] produces the Riordan arrays: C = {[, sn(z, )], [cos(z), sin(z)], [, tanh(z)] respectively The production matrices from B in terms of m =, 0, corresponding to the Riordan arrays in C respectively are as follows: D = , , Remark: The production matrix in D at m = is tridiagonal which indicates that the inverse of [cn(z, ), sn(z, )] = [, tanh(z)] = [, tanh (z) ] is the coefficient array of a family of orthogonal polynomials The generating functions of the r and c sequences associated to the production matrices of B and D are listed below: r(z, m) = cn ( sn (z m) ) ( m dn sn (z m) ) m r(z, ) = z r(z, 0) = z c(z, m) = (m ) ( mz ) cd ( sn (z m) ) ( m nd sn (z m) ) ( m sd sn (z m) ) m c(z, ) = 0 c(z, 0) = z z z 33 [cd(z, m), sc(z, m)] The coefficient matrix of [cd(z, m), sc(z, m)] is given by: 6

65 m A = 0 m m 6m + 0 (m + ) m 6m m 3 07m + 47m 0 3m 46m (m 5) 0 The row sums of A for m = 0 form the sequence M = (,, 0, 0, 4,, 56, ) which has egf cos(z)e tan(z) If M is multiplied with n such that n = 0,,, 3, is the index of M then ( ) n M corresponds to A0094 with egf cos(z) e tan(z) The row sums of A for m = form the sequence (,,,, 5,, 37, ) which corresponds to A00374 with egf e sinh(z) The production matrix of A in terms of m is given by m B = 0 m (m )m m(3m 4) 0 8 m 0 5m ( 3m 7m + 4 ) 0 5(m )m 0 5(m 3) 0 If m =, 0, then [cd(z, m), sc(z, m] produces the Riordan arrays: C = {[cd(z, ), sc(z, )], [cos(z), tan(z)], [, sinh(z)]} respectively The production matrix of C for the case m =, 0, is given by D = , , The r and c generating functions of the production matrix of B and D are listed as follows: r(z, m) = ( z + ) dn ( sc (z m) m ) r(z, ) = z + c(z, m) = (m )cd ( sc (z m) m ) nd ( sc (z m) m ) sd ( sc (z m) m ) (m )z c(z, ) = 0 6

66 34 [cd(z, m), sd(z, m)] The coefficient matrix of [cd(z, m), sd(z, m)] is given by A = m ( 5m 6 ) m 6m + 0 ( 7m 6 ) m 76m ( 3m ) 0 which is equivalent to m m m 6m + 0 (7m 5) m 76m m 0 0 Remark The numbers, 6,, 0, 30, positioned along the (n+) diagonal of the matrix A corresponds to A809 The row sums of A for m = 0 form the sequence (,, 0, 3, 8, 3, 56, ) which has the egf cos(z)e sin(z) The row sums of A for m = form the sequence (,,,, 5,, 37, ) corresponds to A00374 with egf e sinh(z) The production matrix of A in terms of m is given by B = m m 3 0 3m 3 0 9m 6 0 If m =, 0, then [cd(z, m), sd(z, m)] produces the Riordan arrays: C = {[cd(z, ), sd(z, )], [cos(z), sin(z)], [, sinh(z)]} respectively The production matrices of B corresponding to the Riordan arrays C if m =, 0, respectively are given by D = , ,

67 The r and c generating functions of the production matrices B and D are listed below r(z, m) = cd ( sd (z m) m ) nd ( sd (z m) m ) r(z, ) = z + r(z, 0) = z c(z, m) = (m )zcd ( sd (z m) ) ( m nd sd (z m) ) m (m )z + c(z, ) = 0 c(z, 0) = z z 35 [dn(z, m), sn(z, m)] The coefficient matrix of [dn(z, m), sn(z, m)] is given by m A = 0 6 ( m 3 ) m + 4m 0 ( 5m 6 ) m + 44m ( ) m 0 0 m 3 44m 6m 0 9m + 4m ( 7m 6 ) 3 0 which is equivalent to m m m(m + 4) 0 (5m + ) m + 44m + 0 0(m + ) 0 0 m ( m + 44m + 6 ) 0 9m + 4m (7m + 4) 0 Remark The numbers, 6,, 0, 30, positioned along the n +, n diagonal of the matrix A corresponds to A809 The row sums of A for m = 0 form the sequence (,,, 0, 3, 8, 3, ) which correspond to A0007 with egf e sin(z) The row sums of A for m = form the sequence (,, 0, 4, 8, 3, 6, ) which corresponds to A00965 with egf e tanh(z) cosh(z) 64

68 The production matrix of A in terms of m is given by: m B = 0 3m (m )m 0 6m ( 5m 6m + ) 0 (5m + 3) 0 5m ( 3m + m + ) 0 5 ( 3m 4m + ) 0 5(3m + ) 0 If m =, 0, then [dn(z, m), sn(z, m)] produces the Riordan arrays C = {[dn(z, ), sn(z, )], [, sin(z)], [sech(z), tanh(z)]} respectively The production matrices of B associated to the Riordan arrays in C for m =, 0, are as follows: D = , , Remark: The tri-diagonal matrix for m = in D indicates that the inverse of the Riordan array [sech(z), tanh(z)] forms the coefficient array of a family of orthogonal polynomials [ This is A06054 ] the number of degree n permutations with k odd cycles, z tanh (z) is the coefficient array of a family of orthogonal polynomials The three term recurrence relation for the family of orthogonal polynomials is given by P n+ (z) = zp n (z) + n P n (z), n with P 0 (z) =, P (z) = z In particular, let Q n (z) = P n(iz) i n, (i = ) we get Q n+ (z) = zq n (z) n Q n (z), n The generating functions of the r and c sequences associated to the produc- 65

69 tion matrices of B and D are listed below: r(z, m) = cn ( sn (z, m) m ) dn ( sn (z m) m ) = ( z )( mz ) = ( z )( mz ) r(z, ) = z r(z, 0) = z c(z, m) = msn( sn(z, m), m)cn( sn(z, m), m) dn( sn(z, m), m) c(z, ) = z c(z, 0) = 0 = mz z mz 36 [dc(z, m), sn(z, m)] The coefficient matrix of [dc(z, m), sn(z, m)] : m A = 0 6 ( 3 ) m m 6m ( 6 ) 5m ( m m + ) 0 0m 0 0 m m 07m m + 4m ( 7m 6 ) 6 0 which is equivalent to m A = 0 4m m 6m m ( m m + ) 0 0m 0 0 m m 07m m + 4m (7m + ) 0 Remark The numbers, 6,, 0, 30, positioned along the n +, n diagonal of the matrix A correspond to A809 The row sums of A for m = 0 form the sequence (,,, 3, 8, 7, 88, ) corresponding to A00907 with egf 66 e sin(z) cos(z)

70 The row sums of A for m = form the sequence (,,,, 7, 3, 97, ) corresponding to A00373 having egf e tanh(z) The production matrix of [dc(z, m), sn(z, m)] in terms of m is given by m B = 0 3m m 0 6m m + 6m (5m + ) 0 5 ( 3m 3 m + m 3 ) 0 45m + 30m (3m + ) 0 If m =, 0, then [dc(z, m), sn(z, m)] produces the Riordan arrays C = {[dc(z, ), sn(z )], [sec(z), sin(z)], [, tanh(z)], } respectively The production matrices of B associated to the Riordan arrays C if m =, 0, are as follows respectively: D = , , Remark: The tri-diagonal matrix for m = in D indicates that the inverse of the Riordan array [, tanh(z)] forms the coefficient array of a family of orthogonal polynomial sequence This is A594 the triangle of arctanh numbers The r and c sequences corresponding of the production matrices of B and D are listed as follows: r(z, m) = cn ( sn (z m) m ) dn ( sn (z m) m ) r(z, ) = z r(z, 0) = z c(z, m) = (m )zcn ( sn (z m) ) ( m dn sn (z m) ) m (z ) (mz ) c(z, ) = c(z, 0) = z z 37 [cn(z, m), am(z, m)] The coefficient matrix of [cn(z, m), am(z, m)] is given by: 67

71 A = 0 6 ( m 6 ) ( m 6 + ) 4 0 ( m 3 ) m + 34m ( m ) 0 0 6m 44m 0 6m + 44m ( m 3 ) 0 which is equivalent to m m + 0 4m m + 34m (m + ) 0 0 6m 44m 0 6m + 44m (4m + 3) 0 Remark The numbers, 6,, 0, 30, positioned along the n +, n diagonal of the matrix A correspond to A809 The row sums of A for m = 0 form the sequence (,, 0,, 4, 4, 0, ) correspond to A46559 with ogf e z cos(z) The row sums of A for m = form the sequence (,, 0, 3, 4,, 80, ) corresponds to A03 with egf e (sin (tanh(z))) = e gdz (Gudermannian function) The production matrix of A in terms of m is given by B = 0 m m 0 3(m + ) m + 6m 8 0 6m 4 0 5m ( 3m 0m + 4 ) 0 5(m + ) 0 If m = 0, then [cn(z, m), am(z, m)] produces the Riordan arrays: {[cos(z), z], [ sech(z), sin (tanh(z)) ] } The matrix [ sech(z), sin (tanh(z)) ] corresponds to A47308 The inverse matrix of [ sech(z), sin (tanh(z)) ] is [sec(z), log(sec(z) + tan(z))] which corresponds to A47309 Its row sums are the once shifted Euler up/down numbers 68

72 (A000) respectively as follows: D = The production matrices derived from B if m =, 0, are , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = dn(f (z m) m) r(z, ) = sech(f (z )) dn(f (z m) m)sn(f (z m) m) c(z, m) = cn(f (z m) m) c(z, ) = tanh(f (z )) c(z, 0) = tan(z) 3 Appell subgroup of Jacobi Riordan arrays Recall from (4) that the elements of the Appell subgroup are of the form [d(z), z] 3 [cd(z, m), z] The coefficient matrix of [cd(z, m), z] is given by m A = 0 3(m ) m 6m + 0 6(m ) ( 5m 6m + ) 0 0(m ) 0 0 6m 3 07m + 47m 0 5 ( 5m 6m + ) 0 5(m ) 0 Remark The numbers, 3, 6, 0, 5, positioned along the n +, n diagonal of the matrix A correspond to A0007 The row sums of A for m = 0 form the sequence (,, 0,, 4, 4, 0, ) corresponds to A46559 with egf e z cos(z) 69

73 The row sums of A for m = form the sequence (,,,,,,, ) corresponds to A0000 with egf e z The production matrix of A in terms of m is given by m B = 0 (m ) ( m ) 0 3(m ) ( m ) 0 4(m ) 0 6 ( m 3 m + m ) 0 0 ( m ) 0 5(m ) 0 If m =, 0, then [cd(z, m), z] produces the Riordan arrays: C = {[cd(z, ), z], [cos(z), z], [, z]} respectively The production matrices from B if m =, 0, are as follows: D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D and to the Riordan arrays in C are listed below r(z, m) = (m) c(z, m) = (m )nd(z m)sd(z m) cd(z m) c(z, ) = 0 c(z, 0) = tan(z) 3 [nd(z, m), z] The case of the exponential Riordan array [nd(z, m), z] has the coefficient matrix given by m A = 0 3m m(5m 4) 0 6m m(5m 4) 0 0m 0 0 m ( 6m 76m + 6 ) 0 5m(5m 4) 0 5m 0 70

74 Remark The numbers, 3, 6, 0, 5, positioned along the n +, n diagonal of the matrix A correspond to A0007 The row sums of A for m = 0 form the sequence (,,,,,,, ) corresponds to A0000 with egf e z The row sums of A for m = form the sequence (,,, 4, 8, 6, 3, ) corresponds to A078 with egf cosh(z)e z = ez + The production matrix of A in terms of m is given by m B = 0 m (m )m 0 3m (m )m 0 4m 0 6m ( m m + ) 0 0(m )m 0 5m 0 The production matrices for the case m =, 0, in B corresponding to the Riordan arrays C = {[nd(z ), z], [, z], [cosh(z), z]} respectively, are given by: D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = m c(z, m) = mcd(z m)sd(z m) nd(z m) c(z, ) = tanh(z) c(z, 0) = 0 7

75 33 [nc(z, m), z] The coefficient array of [nc(z, m), z] is given by A = m m m 76m m Remark: The numbers, 3, 6, 0, 5, positioned along the n +, n diagonal of the matrix A correspond to A0007 The row sums of A for m = 0 form the sequence (,,, 4,, 36, 5, ) corresponds to A00370 with egf e z cos(z) The row sums of A for m = form the sequence (,,, 4, 8, 6, 3, ) corresponds to A078 with egf cosh(z)e z = ez + The production matrix of A in terms of m is B = m m ( m m + ) m The Riordan matrices from [nc(z, m), z] for m =, 0, are C = {[, nc(z, )], [sec(z), z], [cosh(z), z]} respectively The production matrix from B if m =, 0, are as follows: D = , ,

76 The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = m c(z, m) = dn(z m)sn(z m) cn(z m) c(z, ) = tanh(z) c(z, 0) = tan(z) 33 The Associated subgroup of Jacobi Riordan arrays Recall from (4) that the elements of the Associated subgroup are of the form [, f(z)] 33 [, sn(z, m)] The coefficient matrix of [, sn(z, m)] is given by A = 0 m (m + ) m + 4m + 0 0(m + ) ( m + 3m + ) 0 0(m + ) 0 Remark: The columns of matrix A after the first column are Palindromic The row sums of A for m = 0 form the sequence (,,, 0, 3, 8, 3, ) corresponds to A0007 with egf e sin(z) The row sums of A for m = form the sequence (,,,, 7, 3, 97, ) corresponds to A00373 with egf e tanh(z) The production matrix of A in terms of m is given by B = 0 m (m + ) (m ) 0 6(m + ) (m ) 0 0(m + ) 0 73

77 If m =, 0, then [, sn(z, m)] produces the Riordan arrays: C = {[, sn(z, )], [, sin(z)], [, tanh(z)]} The production matrices from B if m =, 0, are as follows: D = , , The generating functions of the r and c sequences corresponding to the production matrix of B and D are listed as follows: r(z, m) = cn ( sn (z m) ) ( m dn sn (z m) ) m r(z, ) = z r(z, 0) = z c(z, m) = 0 m 33 [, z ns(z, m) ] The exponential Riordan matrix of [, z ns(z, m) ] is given by A = 0 m ( 0 4(m + ) m m + 7 ) 0 0(m + ) ( m m + ) 0 0(m + ) 0 The production matrix of A in terms of m is given by B = 0 m ( 0 3(m + ) m 46m 5 ) ( 0 6(m + ) m + 46m + 5 ) 0 0(m + ) 0 The production matrix B for the case m =, 0, corresponding to the Riordan arrays C = { [, z ns(z ) ], [, z csc(z) ], [, z coth(z) ] }, 74

78 are given by: D = , , [, znc(z, m)] The exponential Riordan matrix of [, znc(z, m)] is given by A = m (m ) Remark The row sums of A for m = 0 form the sequence (,,, 4, 3, 56, 30, ) corresponds to A with egf e z cos(z) The row sums of A for m = form the sequence (,,, 4, 3, 36, 8, ) corresponds to A00377 with egf e z cosh(z) The production matrix of A in terms of m is given by B = m (0m + ) The production matrix above for the case m =, 0, of B corresponding to the Riordan arrays C = {[, znc(z )], [, z sec(z)], [, z cosh(z)]} respectively are given by: D = , ,

79 34 Derivative Subgroup of Jacobi Riordan arrays Recall from (4) that the elements of the derivative subgroup have the form (h (z), h(z)), where h (z) denotes the first derivative of h(z) 34 [ d dz sn(z, m), sn(z, m)] The coefficient array of [ d dz sn(z, m), sn(z, m)] where d dz sn(z, m) = cn(z m)dn(z m) is given by m A = 0 4(m + ) m + 4m + 0 0(m + ) ( m + 3m + ) 0 0(m + ) 0 0 m 3 35m 35m 0 9m + 434m (m + ) 0 Remark: The coefficient matrix of [ d dz sn(z, m), sn(z, m)] forms a Palindromic Riordan array The row sums of A for m = 0 form the sequence (,, 0, 3, 8, 3, 56, ) which has the egf cos(z)e sin(z) The row sums of A for m = form the sequence (,,, 7, 3, 97, 75, ) which has the egf sech (z)e tanh(z) The non-zero entries of the first column of A corresponds to the matrix A[] = By multiplying the matrix A[] by m if n m(mod) where n is the column number st n = 0,,, we get the matrix B[] = The matrix B[] corresponds to A

80 The production matrix of A in terms of m is given by m B = 0 3(m + ) (m ) 0 6(m + ) (m ) 0 0(m + ) 0 45(m ) (m + ) 0 45(m ) 0 5(m + ) 0 If m =, 0, then [ d dz sn(z, m), sn(z, m)] produces the Riordan arrays C = {[cn(z, )dn(z, ), sn(z, )], [cos(z), sin(z)], [ sech (z), tanh(z) ] } respectively The production matrices of B in terms of m =, 0, are as follows: D = , , REMARK: The tri-diagonal production matrix for m = in D which is associated to the Riordan array [ sech (z), tanh(z) ] in C forms an orthogonal polynomial sequence for [ sech (z), tanh(z) ] Furthermore, [ sech (z), tanh(z) ] [ ] = z, tanh (z) represents the coefficient matrix of the family of formal orthogonal polynomials The three term recurrence relation for these polynomials is given by P n+ (z) = zp n (z) + n(n + )P n (z) with P 0 (z) =, P (z) = z st < z < In particular, let Q n (z) = P n(iz) i n, (i = ) we get Q n+ (z) = zq n (z) n(n + )Q n (z), n The generating functions of the r and c sequences corresponding to the pro- 77

81 duction matrix of B and D are listed as follows: A(z, m) = cn ( sn (z m) m ) dn ( sn (z m) m ) = ( sn ( sn(z, m), m)( m sn ( sn(z, m), m)) = ( z )( m z ) A(z, ) = z A(z, 0) = z Z(z, m) = z ( m ( z ) ) cn ( sn (z m) m ) dn ( sn (z m) m ) (z ) (mz ) = z ( m ( z ) ) ( sn ( sn(z, m), m)( m sn ( sn(z, m), m)) (z ) (mz ) = z ( m ( z ) ) ( z )( m z ) (z ) (mz ) Z(z, ) = z ( z ) (z ) = z Z(z, 0) = z z 34 [ d dz sc(z, m), sc(z, m)] The coefficient array of [ d dz sc(z, m), sc(z, m)] where d dz sc(z, m) = dc(z m)nc(z m) is given by Remark: A = m m 0 0 m 6m (m ) 0 The row sums of A for m = 0 form the sequence (,, 3, 9, 37, 77, 959, ) which has the egf sec(z) e tan(z) The row sums of A for m = form the sequence (,,, 5,, 37, 8, ) which has the egf cosh(z)e sinh(z) The production matrix of A in terms of m is given by m B = 0 6 3m 0 0 3m 0 6m 0 0 5m 0 0 0m 0 78

82 If m =, 0, then [ d dz sc(z, m), sc(z, m)] produces the Riordan arrays: C = {[dc(z, )nc(z, ), sc(z, )], [ sec (z), tan(z) ], [cosh(z), sinh(z)]} respectively The production matrices from B corresponding to m =, 0, are as follows D = , , The generating functions of the r and c sequences corresponding to the production matrix of B and D are listed as follows: r(z, m) = ( z + ) dn ( sc (z m) m ) r(z, ) = z + r(z, 0) = z + c(z, m) = z ( (m )z + m ) dn ( sc (z m) m ) (m )z z c(z, ) = z + c(z, 0) = z 343 [ d dz am(z, m), am(z, m)] The coefficient array [ d dz am(z, m), am(z, m)] where d dz am(z, m) = dn(z m) is given by m A = 0 4m m(m + 4) 0 0m m(m + 3) 0 0m 0 0 m ( m + 44m + 6 ) 0 7m(3m + ) 0 35m 0 Remark: The row sums of A for m = 0 form the sequence (,,,,,,, ) corresponds to A0000 with egf e z The row sums of A for m = form the sequence (,, 0, 3, 4,, 80, ) corresponds to A03 with egf e sin (tanh(z)) = e gdz where g is the 79

83 Gudermannian function such that gd(z) = z 0 dt < z < cosh t The non-zero elements of the first column of the matrix A generated from the derivative of the Jacobi amplitude function forms the coefficient matrix A[] = The production matrix of A in terms of m: m B = 0 3m (4 3m)m 0 6m (4 3m)m 0 0m 0 m ( 45m + 60m 6 ) 0 5(4 3m)m 0 5m 0 If m = and m = 0 then [ d dz am(z, m), am(z, m)] produces the Riordan arrays Remark: We note that [ e z C = { e z +, tan (e z ) π ], [, z]} respectively e z e z + + has an ordinary generating function given by + + x 4x 9x + 5x + The production matrices from B in terms of m =, 0, are as follows D = , ,

84 Remark: We note that the generating function of the matrix D at m = is e xy ( sin(x) + y cos(x)) The generating functions of the r and c sequences corresponding to the production matrix of B and D are listed as follows: r(z, m) = dn(f (z m) m) r(z, ) = sech(f (z )) = sech(log(tan(z) + sec(z))) (tan(z) + sec(z)) = (tan(z) + sec(z)) + = cos(z) r(z, 0) = c(z, m) = mcn(f (z m) m)sn(f (z m) m) dn(f (z m) m) c(z, ) = tanh(f (z )) c(z, 0) = 0 = tanh(log(tan(z) + sec(z))) (tan(z) + sec(z)) = + (tan(z) + sec(z)) = sin(z) Recall: The incomplete elliptic integral of the first kind () denoted F (φ, m) where 0 < φ < π If φ = π we have the complete elliptic integral of the first kind K It has a series expansion given by F (φ, m) = φ+ mφ ( 9m 4m ) 5m φ +( m + 6m ) φ 7 +O ( φ 9) 5040 The Mathematica code to compute F (z ) is as follows: EllipticF[z, ] EllipticF[z, ] Pi] FunctionExpand [ ] %, 0 < z < Log[Sec[z] + Tan[z]] 8

85 344 [ d dz sd(z, m), sd(z, m)] The coefficient array of [ d dz sd(z, m), sd(z, m)] where d dz sd(z, m) = cd(z m)nd(z m) is given by A = ( ) m ( 4m 3 ) m 6m + 0 ( 5m 3 ) ( 7m 7m + ) 0 0(m ) 0 which is equivalent to m m m 6m + 0 0m ( 7m 7m + ) 0 40m 0 0 REMARK: The numbers, 6,, 0, 30, 4, located on the n +, n diagonal correspond to A00378 The row sums of A for m = 0 form the sequence (,, 0, 3, 8, 3, 56, ) with egf cos(z)e sin(z) The row sums of A for m = form the sequence (,,, 5,, 37, 8, ) which has egf cosh(z)e sinh(z) If m = and m = 0 then [ d dz sd(z, m), sd(z, m)] produces the Riordan arrays C = {[cosh(z), sinh(z)], [cos(z), sin(z)]} The production matrix of A in terms of m is given by m B = 0 6m m m m m 5 0 The production matrices from B in terms of m =, 0, are as follows: D = , ,

86 The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cd ( sd (z m) m ) nd ( sd (z m) m ) r(z, ) = z + r(z, 0) = z c(z, m) = z ( (m )mz + m ) cn ( sd (z m) ) m (m )z + c(z, ) = z z + c(z, 0) = z z 345 [ d dz sc (z, m), sc (z, m) ] The coefficient matrix of [ d dz sc (z, m), sc (z, m) ] where d dz sc (z, m) = nd( sc (z m) m) z + : A = m (m ) ( 3m 8m + 8 ) 0 0(m ) 0 Remark: The row sums of A for m = 0 form the sequence (,,, 7, 5, 45, 5, ) which has the egf +z e tan (z) The production matrix of A in terms of m is given by B = m (m ) 0 0 m(5m 8) (m ) 0 0 5(m(5m 8) + 8) 0 0(m ) 0 If m = and m = 0 then [ d dz sc (z, m), sc (z, m) ] [ ] [ ] produces the Riordan arrays C = {, z + sinh (z), z +, tan (z) } The production matrices from B for m =, 0, are as follows: D = , ,

87 The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cn(z m) nd ( sc (sc(z m) m) m ) r(z, ) = cosh(z) r(z, 0) = cos (z) c(z, ) = tanh(z)( sech(z)) c(z, 0) = sin(z) c(z, m) = cn(z m) 4 ( sc(z m) + ) (mcd ( sc (sc(z m) m) m ) sd ( sc (sc(z m) m) m ) sc(z m)) 346 [ d dz sd (z m), sd (z m) ] The coefficient matrix [ d dz sd (z m), sd (z m) ] where which is equivalent to d dz sd (z m) = cn ( sd (z m) ) m (m )z A = ( m) ( 3 ) 4m ( ) m m ( 5 6 ) 5m m m ( 8m 8m + 3 ) 0 0 0m 0 REMARK: The numbers, 6,, 0, 30, 4, located on the n +, n diagonal correspond to A00378 The row sums of A for m = 0 form the sequence (,,, 5, 0, 85, 50, ) which has egf z esin (z) 84

88 The production matrix of A in terms of m: B = m m 0 8m 8m m 0 If m = and m = 0 then [ d dz sd (z, m), sd (z, m) ] [ ] [ ] produces the Riordan arrays C = { z + sinh (z), z sin (z) } respectively In particular we have that, [ (z)] z +, sinh = [cosh(z), sinh(z)] The production matrices from B for m =, 0, are as follows D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cn ( sd (sd(z m) m) ) m (m )sd(z m) + r(z, ) = cosh(z) r(z, 0) = cos(z) c(z, m) = dn ( sd (sd(z m) m) m ) sn ( sd (sd(z m) m) m ) + (m )sd(z m) (m )sd(z m) + c(z, ) = tanh(z)( sech(z)) c(z, 0) = tan(z) sec(z) 347 [ d dz sn (z, m), sn (z, m) ] The coefficient array of [ d dz sn (z, m), sn (z, m) ] where d dz sn (z, m) = cd ( sn (z, m) ) m z 85

89 is given by A = m (m + ) ( 3m + m + 3 ) 0 0(m + ) ( 8m + 7m + 8 ) 0 0(m + ) 0 REMARK: The numbers, 4, 0, 0, located on the (n +, n) diagonal correspond to A0009 The row sums of A for m = 0 form the sequence (,,, 5, 0, 85, 50, ) which has egf z esin (z) The row sums of A for m = form the sequence (,, 3, 9, 45, 5, 575, ) which has egf +z z z The production matrix of A expressed in terms of m m B = 0 3(m + ) 0 0 5m m (m + ) ( 5m m + 5 ) 0 0(m + ) 0 If m = and m = 0 then [ d dz sn (z, m), sn (z, m) ] [ ] [ ] produces the Riordan arrays C = z, tanh (z),, z sin (z) respectively In particular, [ ] z, tanh (z) = [ sech (z), tanh(z) ] corresponds ( ) to a signed version of A05949 We recall that tanh (z) = ln +z z The production matrices from B in terms of m =, 0, are as follows: D = , , The generating functions of the r and c sequence corresponding to the produc- 86

90 tion matrices of B and D are listed as follows: r(z, m) = cd ( sn (sn(z m) m) m ) cn(z m) r(z, ) = cosh (z) r(z, 0) = cos (z) ( ) c(z, m) = m dn(z m) + sn(z m) cn(z m) c(z, ) = sinh(z) c(z, 0) = tan(z) sec(z) 4 Products of Jacobi Riordan arrays We use the multiplication rule of Riordan arrays (3) in the context of Riordan arrays defined by Jacobi elliptic functions 4 [cn(z, m), sn(z, m)] Consider the product [cn(z, m), sn(z, m)] = [cn(z, m), sn(z, m)] [cn(z, m), sn(z, m)] [cn(z, m)cn(sn(z,m),m), sn(sn(z, m), m)] The resulting coefficient matrix is given by which is equivalent to A = ( m 3 ) (m + ) 0 ( m 3 ) (m + 4) 0 0 (m + ) 0 4(m + 5) 0 REMARK The numbers, 6,, 0, 30, located on the (n +, n) diagonal of A correspond to A

91 The production matrix of A in terms of m is given by B = (m + 3) 0 8m 4 0 6(m + ) 0 If m = and m = 0 then [cn(z, m), sn(z, m)] produces the Riordan arrays C = {[sech(z)sech(tanh(z)), tanh(tanh(z))], [cos(z) cos(sin(z)), sin(sin(z))]} respectively The production matrices from B in terms of m =, 0, are as follows: D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cn ( sn (z m) m ) cn ( sn ( sn (z m) m ) m ) dn ( sn (z m) m ) dn ( sn ( sn (z m) m ) m ) r(z, ) = ( z ) ( tanh (z) ) r(z, 0) = z sin (z) c(z, ) = tanh (z) ( z tanh (z) ) z c(z, 0) = sin (z) ( z z sin (z) ) + z z sin (z) 4 [cn(z, m), sn(z, m)] [ d sn(z, m), sn(z, m)] dz Consider the product [ ] d [cn(z, m), sn(z, m)] sn(z, m), sn(z, m) = [cn(z, m), sn(z, m)] [cn(z, m)dn(z, m), sn(z, m)] dz = [cn(z, m)cn(sn(z, m), m)dn(sn(z, m), m), sn(sn(z, m), m)] 88

92 The resulting coefficient matrix is given by : A = ( m ) ( 5m 6 ) m + 3m + 0 ( 7m 6 ) which is equivalent to m m m + 3m + 0 (7m + 0) 0 REMARK The numbers, 6,, 0, 30, located on the n +, n diagonal of A correspond to A00378 The production matrix of A in terms of m : is given by B = m m 6 0 4m 4 0 9m 0 The production matrices from B in terms of m =, 0, : are as follows D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cn ( sn (z m) m ) cn ( sn ( sn (z m) m ) m ) dn ( sn (z m) m ) dn ( sn ( sn (z m) m ) m ) r(z, ) = ( z ) ( tanh (z) ) r(z, 0) = z sin (z) 89

93 c(z, ) = z ( z tanh (z) ) ( tanh (z) ) ( ) tanh z tanh (z) c(z, 0) = sin (z) ( z z sin (z) ) + z z sin (z) tanh (z) 43 [ d dz sn(z, m), sn(z, m)] Consider the product: [ ] [ ] [ ] d d d sn(z, m), sn(z, m) = sn(z, m), sn(z, m) sn(z, m), sn(z, m) dz dz dz = [cn(z,m)dn(z,m), sn(z, m)] [cn(z,m)dn(z,m), sn(z, m)] = [cn(z,m)dn(z,m)cn(sn(z,m),m)dn(sn(z,m),m), sn(sn(z, m), m)] The resulting coefficient matrix is given by A = ( m ) (m + ) 0 0 ( m + 4m + ) 0 0(m + ) 0 The production matrix of A in terms of m is given by B = (m + ) (m + ) 0 4 ( m 4m + ) 0 (m + ) 0 If m = and m = 0 then [ d dz sn(z, m), sn(z, m)] produces the Riordan arrays { [ sech (z)sech (tanh(z)), tanh(tanh(z)) ], [cos(z) cos(sin(z)), sin(sin(z))]} respectively The production matrices from B in terms of m =, 0, are as follows: D = , , The generating functions of the r and c sequences corresponding to the produc- 90

94 tion matrices of B and D are listed as follows: r(z, m) = cn ( sn (z m) m ) cn ( sn ( sn (z m) m ) m ) dn ( sn (z m) m ) dn ( sn ( sn (z m) m ) m ) r(z, ) = ( z ) ( tanh (z) ) r(z, 0) = z sin (z) c(z, ) = tanh (z) ( z tanh (z) ) z c(z, 0) = sin (z) ( z z sin (z) ) + z z sin (z) 44 [ d sc(z, m), sc(z, m)] [ d sn(z, m), sn(z, m)] dz dz Consider the product [ ] [ ] d d sc(z, m), sc(z, m) sn(z, m), sn(z, m) dz dz = [dc(z,m)nc(z,m), sn(z, m)] [cn(z,m)dn(z,m), sn(z, m)] = [dc(z,m)nc(z,m)cn(sc(z,m),m)dn(sc(z,m),m), sn(sc(z, m), m)] The resulting coefficient matrix is given by A = ( m) ( ) m 3 3 ( 4m 4m ) 0 ( 5 6 5m 3 ) 0 which is equivalent to m m ( 4m 4m ) 0 0 0m 0 REMARK The numbers, 6,, 0, 30, located on the n +, n diagonal correspond to A

95 The production matrix of A in terms of m is given by B = m m 0 4m + 4m m 0 The production matrices from B in terms of m =, 0, are as follows: D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = ( sn (z m) + ) cn ( sn (z m) m ) dn ( sn (z m) m ) dn ( sc ( sn (z m) m ) m ) r(z, ) = ( z ) tanh (z) + r(z, 0) = z ( sin (z) + ) c(z, m) = c(z, ) = z z tanh (z) + tanh (z) tanh (z) + c(z, 0) = z sin (z) z ( sin (z) + ) z 45 [ d sd(z, m), sd(z, m)] [ d sn(z, m), sn(z, m)] dz dz Consider the product [ ] [ ] d d sd(z, m), sd(z, m) sn(z, m), sn(z, m) dz dz = [cd(z,m)nd(z,m), sd(z, m)] [cn(z,m)dn(z,m), sn(z, m)] = [cd(z,m)nd(z,m)cn(sd(z,m),m)dn(sd(z,m),m), sd(sd(z, m), m)] 9

96 The resulting coefficient array is given by : A = m (m ) ( m + 4m 4 ) 0 0(m ) 0 The production matrix of A in terms of m is given by B = m (m ) 0 7m + 4m 4 0 6(m ) 0 The production matrices from B in terms of m = {, 0, } are as follows: D = , , The generating functions of the r and c sequences corresponding to the production matrices of B and D are listed as follows: r(z, m) = cn ( sn (z m) m ) dn ( sn (z m) m ) cd ( sd ( sn (z m) m ) m ) nd ( sd ( sn (z m) m ) m ) r(z, ) = ( z ) tanh (z) + r(z, 0) = z sin (z) c(z, ) = z z tanh (z) + tanh (z) tanh (z) + c(z, 0) = z sin (z) + z z sin (z) z sin (z) 93

97 46 [ d dz sc (z, m), sc (z, m) ] [ d sn(z, m), sn(z, m)] dz For the case of [ d dz sc (z, m), sc (z, m) ] [ d dz sn(z, m), sn(z, m)] its coefficient array is given by A = 0 3(m + 3) (m + 3) ( m + m + 5 ) 0 30(m + ) ( 8m + m + 5 ) 0 5(4m + 3) 0 The production matrix of A in terms of m is given by B = 0 3(m + ) (m ) 0 9(m + ) ( m 4m + 8 ) 0 6(3m + ) 0 45 ( 3m m + 8 ) 0 45 ( m m + 4 ) 0 5(m + ) 0 The production matrices of B interms of m =, 0, are respectively given by , , Jacobi Riordan arrays of complex variables with imaginary modulus k In this section we construct Riordan arrayus using Jacobi elliptic functions of a complex variable u with an imaginary modulus k 5 The Riordan array from sn(iu, k ) The imaginary transformation of the elliptic sinus is sn(iu, k ) = i sn(u,k) cn(u,k) having series expansion given by sn(u, k) i cn(u, k) = iu 6 i(k )u3 + 0 i ( k 6k + 6 ) u 5 i ( k 3 38k + 408k 7 ) u O ( u 9) (3) 94

98 5 [ ], i sn(u,k) cn(u,k) The [ exponential ] Riordan array of the Lagrange subgroup corresponding to (3) is, i sn(u,k) cn(u,k) having coefficient matrix given by i A = 0 i(k ) 0 i (k ) i ( k 6k + 6 ) 0 0i(k ) 0 i ( k + 7k 7 ) 0 0(k ) 0 The production matrix of A in terms of k is given by 0 i i B = 0 i(k ) 0 i i(k ) 0 i 0 0 3ik 0 6i(k ) 0 i 0 0 5ik 0 0i(k ) 0 The production matrix of B for the cases k =, 0, is given by 0 i i i 0 i i 0 i 0 0 3i 0 8i 0 i 0 0 5i 0 30i 0, 0 i i i 0 i i 0 i i 0 i i 0, 0 i i i 0 i i 0 i 0 0 3i 0 6i 0 i 0 0 5i 0 0i 0 5 [ d du ( i sn(u,m) cn(u,m) ) ], i sn(u,m) cn(u,m) The exponential [ ( Riordan ) array ] of the derivative subgroup corresponding to d (3) is du i sn(u,m) cn(u,m), i sn(u,m) cn(u,m) [ d du sn(iu, k ), sn(iu, k ) ] having coefficient matrix given by i A = i(m ) 0 i (m ) 0 0 i ( m 6m + 6 ) 0 0i(m ) 0 i The production matrix of A in terms of m is given by B = 0 i 0 0 i(m ) 0 i 0 0 3i(m ) 0 i 3im 0 6i(m ) 0 95

99 The production matrices from B in terms of m are as follows: D = 0 i 0 0 3i 0 i 0 0 9i 0 i 3i 0 8i 0, 0 i 0 0 i 0 i 0 0 6i 0 i 0 0 i 0, 0 i 0 0 i 0 i 0 0 3i 0 i 3i 0 6i 0 5 The Riordan array from cn(iu, k ) The transformation of the imaginary cosine is given by cn(iu, k ) = which has the series expansion given by nc(u, k) cn(u, k) ( cn(u, k) = + u k ) u 4 + ( 6k 76k + 6 ) u ( 64k k 44k ) u 8 + O ( u 9) (4) 4030 The exponential Riordan array corresponding to (4) of the the Appell subgroup defined by [ ] cn(u, k), u is given by REMARK A = k k k 76k k The row sums for case k = 0 is given by {,,, 4,, 36, 5, } corresponds to A00370 and having egf e u cos(u) The row sums for the case k = is given by {,,, 4, 8, 6, 3, } corresponds to A078 and has the egf cosh(u)e u = eu + 96

100 The production matrix of A in terms of k is given by B = k k ( k k + ) k The production matrix B for the cases k =, 0, is given by , , The Riordan array from dn(iu, k ) The imaginary transformation of elliptic dn is dn(iu, k ) = and has the series expansion given by dn(u, k) cn(u, k) dc(u, k) ( + k ) u + ( k 6k + 5 ) u 4 + ( k k 07k + 6 ) u ( k 4 4k 3 + 4k 36k ) u 8 + O ( u 9) (5) 4030 The exponential Riordan array corresponding to (5) is defined by and has the coefficient matrix given by [ ] dn(u, m) cn(u, m), u [dn(iu, k), iu] A = ( ) m ( ) m 0 0 m 6m ( ) m 0 97

101 which is equivalent to A = m m 0 0 m 6m m 0 Remark: The sequence of numbers, 6,, which are located on the n +, n diagonal of A correspond to A00378 The row sums of A form the sequence (,,, 4,, 36, 5, ) corresponds to A00370 with egf e u cos(u) The row sums of A forming the sequence (,,,,,,,, ) correspond to A0000 with egf e u The production matrix of A in terms of m is given by B = m m 0 m 0 3 3m 0 The production matrices from B in terms of m =, 0, are as follows: , , The Riordan array from cd(iu, k ) The transformation of the imaginary elliptic Jacobi cd is given by cd(iu, k ) = dn(u,k) nd(u, k) which has the series expansion such that ( dn(u, k) = + ku 5k + 4 k ) u 4 + ( 6k 3 76k + 6k ) u ( 385k 4 44k k 64k ) u 8 + O ( u 9) (6)

102 [ ] The exponential Riordan array corresponding to (6) is dn(u,k), u coefficient matrix given by having k A = 0 3k k(5k 4) 0 6k k(5k 4) 0 0k 0 0 k ( 6k 76k + 6 ) 0 5k(5k 4) 0 5k 0 REMARK: The row sums of A for k = 0 are {,,,,,,, } corresponding to A0000 The row sums for k = given by {,,, 4, 8, 6, 3} corresponds to A078 The production matrix of A in terms of k is given by k B = 0 k (k )k 0 3k (k )k 0 4k 0 6k ( k k + ) 0 0(k )k 0 5k 0 The production matrix of B for k =, 0, is given by , , Jacobi Riordan arrays from the A and Z generating functions In general an exponential Riordan array [g(x), f(x)] can be expressed in terms of its A and Z generating functions such that [g(x), f(x)] = [ e Rev( x 0 0 dt A(t)) Z(t) x A(t) dt, Rev 0 ] dt A(t) 99

103 Example Suppose that A(t) = ( t )( k t ) ( ) x dt Then Rev ( t )( k t ) 0 = sn(x) So f(x) = sn(x) sn(x) Z(t) 0 g(x) = e A(t) dt For instance if Z(t) = k t then we have The coefficient array of sn(x) 0 g(x) = e dt t So [g(x), f(x)] = = e sin (sn(x)) [ ] e sin (sn(x)), sn(x) [ ] e sin (sn(x)), sn(x) is given by A = m m m 8m 4m m 6m + m 6m 4 0(3m + ) 0m 5 0 6m + 4m + 8 ( 4m + 6m ) 6m 6m 9 40(m + ) 5(4m + ) 6 Remark The row sums of A at m = 0 corresponds to A0098 having e,g,f e (z+sin(z)) The production matrix of A is given by B = m m m 3(m + ) 0 3m 3(m ) 6m 6(m + ) 0 5m 5(m ) 0m 0(m + ) 00

104 The production matrix B at m =, 0, is given by , , Similarly if, then we get, Z(t) = t g(x) = e k sin (ksn(x)) So The coefficient matrix of [g(x), f(x)] = [ ] e k sin (ksn(x)), sn(x) [ ] e k sin (ksn(x)), sn(x) is given by A = 0 m (m + ) 4m (m ) m + 4m 4 0(m + ) 0m 5 0 4m 3 6 ( m + 8m 7 ) 6m + 44m 89 60(m + ) 5(4m + ) 6 The production matrix of A in terms of m is given by B = m (m + ) 0 3 3(m ) 6 6(m + ) 0 5 5(m ) 0 0(m + ) The production matrix B if m =, 0, is given by , , Example Let us consider the exponential Riordan array M = [cn(x), sn(x)] 0

105 where we suppress the parameter k We then have f(x) = sn (x) We also have g (x) = cn (x) = sn(x)dn(x), so that we obtain Z(x) = Z M (x) = g ( f(x)) g( f(x) = sn ( sn (x) ) dn ( sn (x) ) cn (sn (x)) = x k x x Thus for the exponential Riordan array [cn(x, k), sn(x, k)] we have A(x) = ( x )( k x ), Z(x) = x k x x This means in particular that the bivariate generating function of the production matrix of [cn(x), sn(x)] is given by ( e xy x k x + y ) ( x )( k x ) x In this case, we have Z(t) A(t) = t k t t ( t )( k t ) or Thus Z(t) A(t) = t t e Rev( x 0 0 dt A(t)) Z(t) A(t) dt sn(x,k) 0 = e t t dt = sn(x, k) = cn(x, k), as expected To calculate the inverse array [cn(x), sn(x)] we have 0

106 and so [cn, sn(x)] = g( f(x)) = cn(sn (x)) =, x [ ] x, dt x ( k t )( t ) 0 Example We next look at the exponential Riordan array We once again have [ ] cn(x) + sn(x), sn(x) A(x) = ( x )( k x ) Now g(x) = cn(x) +sn(x), and we find that g (x) = ( + sn(x))sn(x)dn(x) cn(x) dn(x) ( + sn(x)) We then get g (sn (x) g(sn ) = ( + x)x k x ( x ) k x + x ( + x) x k x = x [ ] Thus the bivariate generating function for the production matrix of cn(x) +sn(x), sn(x) is given by e xy ( k x x + y ( x )( k x ) ) We note that for k =, we get the exponential Riordan array [ ] sech(x) + tanh(x), tanh(x) 03

107 The inverse matrix is calculated as follows [ ] cn(x) + sn(x), sn(x) = = [ cn(sn (x) +sn(sn (x)) + x x, x 0, sn (x) ] dt ( k t )( t ) Example Our next example is the exponential Riordan array [ ] cn(x) + sn(x), sn(x) + sn(x) We do not immediately know what the inverse function of use the theory of Riordan arrays to continue the analysis Thus we note that sn(x) +sn(x) [ ] [ ] cn(x) + sn(x), sn(x) = [cn(x), sn(x)] + sn(x) + x, x, + x is, so we where the second Riordan array in the product is related to the Laguerre polynomials Taking inverses, we obtain or [ ] [ ] cn(x) + sn(x), sn(x) = + sn(x) + x, x [cn(x), sn(x)], + x [ ] [ ] [ cn(x) + sn(x), sn(x) = + sn(x) x, x x This gives us x, x 0 ] dt ( t )( k t ) [ ] [ cn(x) + sn(x), sn(x) =, + sn(x) x Thus in particular, we have that sn(x) + sn(x) = Rev 0 x x 04 0 x x dt ( t )( k t ), ] dt ( t )( k t )

108 or by the change of variable y = t +t, we get sn(x) x + sn(x) = Rev dy 0 ( y)( y (k )y ) We can generalize this to the following We assume given two exponential Riordan arrays, M = [g, f], and N = [u, v] We assume that M is an elliptic Riordan array, with A M (t) = R(t, P (t)) We consider the product M N = [g, f] [u, v] = [gu(f), v(f)] Knowing that x dt f(x) = Rev 0 A(t), we wish to find an elliptic characterisation of v(f) For this, we look at the inverse Now so we obtain Thus we have that (M N) = N M = f(x) = x 0 [ ] [ ] u( v), v, f g( f) dt A(t), [ (M N) = u( v) g( f( v)), v(x) v(x) dt v(f) = Rev 0 A(t) 0 ] dt A(t) To put this in an elliptic form, we use the change of variable in the integral This gives us y = v(t) = t = v(y) 05

109 dy dt = v (t) = dt = dy v (t) = dy v ( v(y)) = v (y)dy When t = v(x), we have y = v(t) = v( v(x)) = x, and so we have Thus we have v(x) 0 dt x A(t) = v (y)dy 0 A( v(y)) x v(f) = Rev 0 v (y)dy x A( v(y)) = Rev 0 dy v ( v(y))a( v(y)) Example In this example, we seek to write the elliptic function sn(x)( + sn(x)) sn(x) as the reversion of an integral whose limits are 0 to x For this, we consider the elliptic Riordan array [[cn(x), sn(x)] ] and the transformation given by the exponential Riordan array Thus we have the product x, x(+x) x Here, [ ] [ ] x( + x) cn(x) sn(x)( + sn(x)) [cn(x), sn(x)], =, x x sn(x) sn(x) [ ] [ x( + x) + 6x + x x, =, x x ] + 6x + x x and In particular, Also, + 6x + x x v(x) =, v (x) = 3 + x + 6x + x + 6x + x x dt sn(x) = Rev 0 ( k t )( t ) 06

110 Thus we have sn(x)( + sn(x)) sn(x) x = Rev 0 3+y +6y+y dy +6y+y ( ( ) ) ( k +6y+y y ( +6y+y y ) ) 07

111 Chapter 3 Weierstrass Elliptic Functions and Riordan Arrays 3 Introduction The previous chapter treated the Jacobi elliptic functions and the construction of Riordan arrays from these functions to form the Jacobi Riordan arrays The Jacobi elliptic functions have a simple analytical relation to the Weierstrass function The link between the Weierstrass function and the Jacobi elliptic function is given explicitly by (z) = where m is the elliptic modulus sn(z, m) m + 3 The main purpose of this chapter is to establish the relationship between the Weierstrass elliptic functions and Riordan arrays forming new lower triangular matrices known as Weierstrass Riordan arrays The relationship between the Weierstrass functions and Riordan arrays can be examined based on the Taylor series expansion of these functions in both the cases of the elliptic modulus m and their invariants {g, g 3 } The origins of the Weierstrass functions as earlier discussed in section (8) 08

112 can be traced back to the 9th century pioneering work of two famous mathematicians NH Abel(87) and K Weierstrass(855,86) The first known Weierstrass function was referred to as the Weierstrass P Further developments of the subject from G Eisentein (847) and K Weierstrass (855,86,895) led to the formulation of the sigma and zeta Weierstrass functions [8] The Weierstrass elliptic function (z) [45] on the complex plane over a period lattice such that (z) = z + m,n Z can be defined by an infinite sum { } (z mω nω ) (mω + nω ), where ω and ω are called the half periods and m, n 0 The periods of are K and ik where K is defined as K but using k = k instead of k Let ω = K and ω = ik, be the half periods of and set e = (ω), e = (ω ), e 3 = (ω ), where ω + ω = ω The Weierstrass elliptic function (z) satisfies the differential equation (z, k) = 4( (z, k) e )( (z, k) e )( (z, k) e 3 ) The Weierstrass elliptic function denoted (z; g, g 3 ) depends on the argument z and the two parameters {g, g 3 } A meromorphic function is considered to be a Weierstrass elliptic function if it is doubly periodic with two periods usually denoted by ω and ω such that (z + ω) = (z + ω ) = (z) and it satisfies the differential equation (z) = 4 3 (z) g (z) g 3 and z = t=w dt 4t3 g t g 3 st w = (z) 09

113 where g = 60 (mω +nω ) 4 & g 3 = 40 (mω +nω ) 6 with the prime ( ) indicating summation over Z excluding (m, n) = (0, 0) [45] Furthermore, the Weierstrass elliptic function and its derivative parameterizes the elliptic curve over complex numbers Figure 3: Plot depicting a Weierstrass over a complex plane with g 3 = + i Figure 3: The plot of Weierstrass elliptic function having {g, g 3 } = {, } The Weierstrass elliptic function is the most basic Weierstrass elliptic function from which the other types of Weierstrass pseudo- elliptic functions can be constructed The other two most important types of Weierstrass pseudo-elliptic functions are the Weierstrass zeta function (ζ) and Weierstrass sigma function (σ) The Weierstrass zeta ζ function is an odd function such that ζ( z) = ζ(z) & ζ (z) = (z) 0

114 The Weierstrass zeta ζ function is quasi-periodic such that [5] ζ(z + ω k ) = ζ(z) + η k, k =,, 3 The Weierstrass sigma function is also an odd function such that σ( z) = σ(z) & σ (z) σ(z) = ζ(z) The Weierstrass σ is quasi-periodic such that σ(z + ω k ) = e (η k(z+ω k ))σ(z) The Weierstrass σ function is connected to the Weierstrass function by the formula σ(x + y)σ(x y) σ (x)σ (y) = (y) (x) The Taylor series expansion around z = 0 of the Weierstrass elliptic functions in terms of g, g 3 are listed below (z) = z + g 0 z + g 3 8 z4 + g 00 z6 + 3g ( ) g 3 g z g 3 z g g z + O ( z 3) (z) = z 3 + g 0 z+g 3 7 z3 + g 00 z5 + 3g ( ) g 3 g z g 3 z 9 + g g z + O ( z 3) ζ(z) = z g 60 z3 g 3 40 z5 g 8400 z7 g ( g z9 + g g ) 3 z + O ( z 3)

115 σ(z) = z g 40 z5 g z7 g 680 z9 g g z + O ( z 3) The invariants g and g 3 can be expressed in terms of m such that we have: g = 4 ( m m + ) 3 g 3 = 4 ( m 3 3m 3m + ) 7 The corresponding Taylor series expansions of the Weierstrass elliptic functions around z = 0 in terms of the modulus m are listed below (z) = z + ( m m + ) z + ( m 3 3m 3 + ) z ( m 4 m 3 + 3m m + ) z 6 + O ( z 8) 675 ζ(z) = ( m m + ) z ( m m + ) z 3 45 ( m 3 3m 3m + ) z z ( σ(z) = z + z z3 6 + z4 8m 4 + 8m 7 ) ( z 5 6m + 6m ) z O ( z 7) 3 Examples of Weierstrass Riordan arrays We provide several coefficient matrices which define Riordan arrays based on the generating functions of Weierstrass functions together with their corresponding production matrices The Riordan arrays are computed seperately in terms of either the invariants {g, g 3 } and the elliptic modulus m The first set of examples below give the Riordan arrays expressed in terms of the invariants 3 Weierstrass Riordan arrays in terms of the Invariants {g, g 3 } The examples below are Weierstrass Riordan arrays in terms of the invariants

116 3 The Riordan array [, σ(z; g, g 3 )] The coefficient array of [, σ(z; g, g 3 )] is given by A = g g g 3 0 g g 3 0 8g The production matrix of A is given by B = 0 g g g 3 0 5g g g The Riordan array [ d dz z3 (z; g, g 3 ), z 3 (z; g, g 3 ) ] The coefficient array of [ d dz z3 (z; g, g 3 ), z 3 (z; g, g 3 ) ] where d dz z3 (z; g, g 3 ) = 3z (z; g, g 3 ) + z 3 (z; g, g 3 ) is given by A = g g g 3 0 6g The production matrix of A is given by B = g g g g

117 33 The Riordan array [ d dz z ζ(z; g, g 3 ), z ζ(z; g, g 3 ) ] The coefficient array of [ d dz z ζ(z; g, g 3 ), z ζ(z; g, g 3 ) ] where d dz z ζ(z; g, g 3 ) = zζ(z; g, g 3 ) z (z; g, g 3 ) results to A = g g g 3 0 4g g 3 0 g g g 3 0 5g The production matrix of A is given by g B = g g g g g g g g The Riordan array [, z ζ(z; g, g 3 ) ] The coefficient array of [, z ζ(z; g, g 3 ) ] is given by A = g g g 3 0 4g g 3 0 g The production matrix of A is given by B = g g g g g g Remark: The diagonal sequence of elements 0,, 0, 30, 70, in B correspond to A

118 35 The Riordan array [ d dz σ(z; g, g 3 ), σ(z; g, g 3 ) ] The coefficient array of [ d dz σ(z; g, g 3 ), σ(z; g, g 3 ) ] where σ(z; g, g 3 )ζ(z; g, g 3 ) is given by A = The production matrix of A is given by B = g g g 3 0 g g 3 0 8g g 4 0 6g g g g g 3 0 5g g g g g g Weierstrass Riordan arrays in terms of the Modulus m In this section the power series expansion of the the 3 main types of Weierstrass functions in terms of m listed in section (3) will be used to define Riordan arrays according to their subgroups 3 Appell Subgroup of Weierstrass Riordan arrays 3 [ z (z, m), z ] The coefficient array in terms of m of [ z (z, m), z ] is given by A = ( m m + ) ( m m + ) ( m 3 3m 3m + ) 0 4 ( m m + ) Remark: The columns of the matrix A are palindromic 5

119 The production matrix of A in terms of m is given by B = ((m )m + ) ((m )m + ) (m )(m + )(m ) 0 6((m )m + ) The production of B in terms of m = {, 0, } is given by C = , , [zζ(z, m), z] The coefficient array in terms of m of [zζ(z, m), z] is given by A = ( m m + ) ( m m + ) ( m 3 + 3m + 3m ) 0 8 ( m m + ) Remark: The columns of the matrix A are palindromic The production matrix of A in terms of m is given by B = ((m )m + ) ((m )m + ) (m )(m + )(m ) ((m )m + ) The production matrix of B in terms of m =, 0, is given by C = , ,

120 34 Lagrange Subgroup of Weierstrass Riordan arrays 35 [, zσ(z, m)] The coefficient array in terms of m for [, zσ(z, m)] is given by A = ( m 8m 7 ) The production matrix of A in terms of m is given by B = ((m )m 79) The production matrix of B in terms of m = 0, is given by C = , [, zζ(z, m)] The coefficient array in terms of m for [, zζ(z, m)] is given by A = ( m m + ) ( m m + ) The production matrix of A in terms of m is given by B = ((m )m + ) ((m )m + )

121 The production matrix of B in terms of m =, 0, is given by C = , , Derivative subgroup of Weierstrass Riordan arrays 38 [ d dz z3 (z, m), z 3 (z, m) ] The coefficient array in terms of m of [ d dz z3 (z, m), z 3 (z, m) ] where ( d dz z3 (z, m) = 3z sn(z m) + ( m ) 3 is given by A = 80 3 ) z3 cn(z m)dn(z m) sn(z m) ( m m + ) ( m m + ) ( m 3 3m 3m + ) 0 68 ( m m + ) It can be noted that the Riordan array [ d dz z3 (z, m), z 3 (z, m) ] produces the following generating functions for the case m = and m = 0 below If m = then d dz z3 (z, m) becomes ( 3z coth (z) ) z 3 coth(z)csch (z) 3 If m = then z 3 (z, m) becomes ( z 3 coth (z) ) 3 If m = 0 then d dz z3 (z, m) becomes ( 3z csc (z) ) z 3 cot(z) csc (z) 3 8

122 If m = 0 then z 3 (z, m) becomes ( z 3 csc (z) ) 3 Remark: The columns of the matrix A are palindromic The production matrix in terms of m is given by B = ((m )m + ) ((m )m + ) (m )(m + )(m ) 0 0((m )m + ) The production matrix of B in terms of m =, 0, C = , , [ d dz zσ(z, m), zσ(z, m)] The coefficient array in terms of m for [ d dz zσ(z, m), zσ(z, m)] is given by A = ( m 8m 7 ) ( m 6m + ) ( 5 m m 9 ) The production matrix of A in terms of m is given by B = ((m )m 79) (m )m ((m )m 79)

123 The production matrix of B in terms of m = 0, is given by C = , [ d dz zζ(z, m), zζ(z, m)] The coefficient array in terms of m for [ d dz zζ(z, m), zζ(z, m)] is given by A = ( m m + ) ( m m + ) ( m 3 + 3m + 3m ) 0 56 ( m m + ) Remark: The columns of the matrix A are palindromic The production matrix of A in terms of m is given by B = ((m )m + ) ((m )m + ) (m )(m + )(m ) 0 40((m )m + ) The production matrices from B for m = {, 0, } is given by C = , ,

124 3 Bell Subgroup of Weierstrass Riordan arrays 3 [σ(z, m), zσ(z, m)] The coefficient array in terms of m of [σ(z, m), zσ(z, m)] is given by A = ( m 8m 7 ) ( m m 9 ) ( 6m + 6m ) 3 5 The production matrix of A in terms of m is B = ((m )m + ) ((m )m + ) 8 5 ((m )m 469) The production matrix of B inn terms of m = 0, is C = , [ z (z, m), z 3 (z, m) ] The coefficient array in terms of m of [ z (z, m), z 3 (z, m) ] is given by A = ( m m + ) ( m m + ) ( m 3 3m 3m + ) 0 7 ( m m + ) Remark: The columns of the matrix A are palindromic

125 The production matrix of A in terms of m is given by B = ((m )m + ) ((m )m + ) (m )(m + )(m ) 0 56((m )m + ) The production matrix of B in terms of m =, 0, is given by C = , ,

126 Chapter 4 Dixonian Elliptic Functions and Riordan arrays 4 Introduction The Dixonian elliptic functions (A0434, A0433) were first introduced in the seminal work of the English mathematician Alfred Cardew Dixon (890)[3] The first significant contribution of the Dixonian elliptic functions in mathematics, was that they parameterized the Fermat cubic curve x 3 + y 3 3axy =, for the case a = 0, for which the functions display a special hexagonal symmetry In particular, this property corresponds to the case for which g = 0 in the Weierstrass theory or the case g 3 = 0 for the lemniscate The two main types of Dixonian elliptic functions are the Dixonian sine sm and the Dixonian cosine cm The Trigonometric Dixonian functions are defined in terms of the and functions such that and ( ) cm(z) = 3 z; 0, ( z; 0, 7) sm(z) = 6 ( ) z; 0, 7 3 ( ) z; 0, 7 3

127 These functions satisfy the non-linear differential system sm (z) = cm(z) & cm (z) = sm(z) such that and sm(0) = 0 cm(0) = sm 3 (z) + cm 3 (z) = The Dixonian elliptic functions sm and cm have periods 3K and 3ωK where ω = + i 3 is a cube root of unity, such that for j = 0,,, sm(z + 3ω j K) = sm z cm(z + 3ω j K) = cm z The Dixonian functions sm and cm as expressed in [6, 65] can explicitly be defined as z sm(z) = Rev 0 ( dt and cm(z) = ( t 3 ) /3 z ) dt (4) ( t 3 ) /3 Furthermore, Conrad & Flajolet [6] also established the combinatorial and probabilistic significance of sm and cm This connection from these two important areas of mathematics is based on the unusual continued fraction expansions derived from their corresponding Laplace transforms 4 Dixonian Riordan arrays In this section we establish the relationship between Dixon elliptic functions and Riordan arrays by constructing proper exponential Riordan arrays based on the Dixon elliptic functions The Taylor series expansion around z = 0 for these functions is given by cm(z) = z3 3 + z6 8 3z z O ( z 3) z3 3! +40z6 6! 3680z9 9! +O ( z 3) 4

128 and sm(z) = z z4 6 +z7 63 3z0 68 +O ( z 3) z 4 z4 4! + 60z7 7! 0800z0 0! + O ( z 3) sm(z) defined in (4) is suited exponential Riordan arrays with A(z) = ( z 3) /3 of the form [ z ] dt g(z), Rev 0 A(t) for suitable Z(t) where g(z) = sm(z) 0 Z(t) A(t) dt 4 The Riordan array [ d dz sm(z), sm(z)] [cm(z), sm(z)] The coefficient array of [ d dz sm(z), sm(z)] is given by M = The production matrix of M is given by P = The A and Z generating functions of P are determined below We know that for an exponential Riordan array [g, f] A(z) = ( z 3) /3 and f(z) = sm (z) 5

129 Now, where So But Z(z) = g ( f) g( f) g(z) = sm (z) = cm (z) g( f) = cm (sm (z)) cm 3 + sm 3 = = cm = ( sm 3) /3 So cm (sm (z)) = [( sm 3 (sm (z)) /3] = ( z 3) /3 = g( f) Also g ( f) = (cm ) (sm (z)) = (cm cm )(sm (z)) = cm(sm (z))( sm (sm (z))) = ( sm 3 (sm (z)) /3 ( z ) = ( z 3 ) /3 ( z ) = g ( f) g( f) = z ( z 3 ) /3 z = ( z 3 ) /3 ( z 3 ) /3 That is Z(z) = z ( z 3 ) /3 and A(z) = ( z 3 ) /3 4 The Riordan array [ d zcm(z), zcm(z)] dz Consider the coefficient array of [ d dz zcm(z), zcm(z)] 6

130 M = The production matrix M is given by: P = The Riordan array [, sm(z)] The proper exponential Riordan array [, sm(z)] of the Lagrange subgroup has the coefficient matrix given by M =

131 The production matrix corresponding to the coefficient matrix of M is given by P = The A generating function corresponding to the production matrix P is given by A(z) = ( z 3 ) /3 44 The Riordan array [cm(z), z] The proper exponential Riordan array [cm(z), z] of the Appell subgroup has the coefficient matrix given by: M = The production matrix corresponding to the coefficient matrix of M is given by P = Let [g(z), f(z)] = [cm(z), z] Then the A generating function of P is given by 8

132 by A(z) = since f (z) = The Z generating function is g ( f) g( f) = cm (z) cm(z) = d dz ln(cm(z)) 9

133 Chapter 5 Embedded Sub-Matrices from a Riordan array 5 Introduction Embedded sub-structures forming new Riordan arrays from existing Riordan arrays have been investigated from different contexts The various research to date on embedded Riordan arrays are briefly presented below Barry [] investigates the concept of embedded Riordan arrays In the case of embedded Riordan arrays, the process of obtaining the embedded Riordan array from an existing Riordan array (d(t), h(t)) involves taking every second column after the first column resulting ( in) another lower triangular matrix A that can be represented as A = d, h z The m complementary Riordan arrays is presented in the publication by Luzón et al [69] The concept of complementary Riordan arrays states that if D = R (d(t), h(t)) then its [m] complementary array is the Riordan array D [m] = ( d( h(t)) h (t) ( ) h(t)) m+ t, h(t) In the r-shifted central coefficients [6], by setting D = (d(z), h(z)) = (d i,j ) i,j 0 to represent a Riordan array then the r-shifted central coefficients results to the new Riordan array D n+r,n+r, where n, r N 0 30

134 The row elimination procedure by Brietzke[9], is a type of embedded Riordan array constructed from a given Riordan array uses a process of eliminating entire rows and some parts of the remaining rows That is if given a proper Riordan array {d n,k } n,k 0, then for any integers p and r 0, d n,k = d pn+r,(p )n+r+k (n, k 0) defines a new Riordan array All these mentioned methods have relied on manipulating the existing generating functions or by shifting the position of the elements that make up the original ordinary Riordan array In this chapter we extend the existing examples of embedded Riordan arrays by identifying some sub-matrices forming new Riordan arrays from an existing exponential Riordan array generated from elliptic Jacobi functions Based on the structure of the elements of Jacobi Riordan arrays and inspired by the existing knowledge on embedded Riordan arrays, some interesting sub-matrices corresponding to the monic polynomials from elliptic Jacobi dc function are examined In addition, to using the already known methods of determining embedded Riordan arrays, two new recent techniques to form new Riordan arrays from existing ones are presented using examples of some elliptic Jacobi functions These are the r-shifted central triangles of a Riordan array connected to the r-shifted central coefficients and the one-parameter family of Riordan arrays derived from a given Riordan array connected to the complementary Riordan arrays The examples presented below using Jacobi elliptic functions illustrate the application of some of these techniques 5 The submatrices of Riordan arrays generated by Elliptic Functions In this section we investigate the submatrices derived from the first or second column of some Riordan arrays generated from elliptic functions The case of the sub-matrices from the first column are associated to their original Riordan matrices belonging to the Appell subgroup On the other hand the submatrices arising from the second column are associated to their original Riordan matrices belonging to the Lagrange subgroup 3

135 [ ] 5 The submatrix of dn(z,m), z cn(z,m) The proper exponential Riordan array by [ ] dn(z,m) cn(z,m), z has coefficient matrix given C = m m m 6m m ( m 6m + 5 ) 0 0(m ) 0 The non-zero elements of the first column of matrix C are associated to the matrix: D = The first column of D forms the sequence (,, 5, 6, 385, ) which are referred to as the Euler numbers having egf sec(z) corresponding to A The row sums of D are {, 0, 0, 0, 0, 0} having the simple generating function z corresponding to A00054 Alternatively, the matrix D can be transformed by multiplying with ( ) n where n is the column number resulting to E = In particular, the row sums of E form the sequence (,,, 6, 7056, 36898, ) corresponding to A5330 with gf A(z) = cm4(z) + sm4(z) where cm4(z) and sm4(z) are the gfs of A53300 and A5330, respectively, that satisfy cm4(z) 4 sm4(z) 4 = 3

136 [ ] 5 The submatrix of dn(z,m), z cn(z,m) The coefficient matrix of the proper exponential Riordan array given by A = m m m 6m m m 80m m 0 [ ] dn(z,m) cn(z,m), z is The non-zero elements of the first column of the matrix A are associated to the matrix: B = The elements of the first column of B forms the sequence (,, 6, 7, ) having Egf tan(z) corresponding to A0008 Furthermore, the row sums of B are given by (,,,,,, ) having egf e z which corresponds to A0000 Using the generating function for the row sums of B given by e z and the generating function of the first column of B given by tan(z) we determine a new Riordan array from these results below Recall that for an exponential Riordan array [g(z), f(z)], the row sums are given by g(z)e f(z) where g(z) F 0 and f(z) F But tan(z) F since tan(z) = z + z3 3 + z z O ( z 9) 33

137 Therefore solving to determine the first generating function of the Riordam array we get That is g(z) = e z tan(z) F 0 since e z = g(z)e tan(z) g(z) = e z e tan(z) = e z tan(z) e z tan(z) = z3 3 z5 5 + z6 8 7z z O ( z 9) The coefficient matrix of the exponential Riordan array [ ] e z tan(z), tan(z) is given by C = The production matrix of C is given by 53 The submatrix of [ ] dn(z,m), z cn(z,m) 3 The coefficient matrix of the proper exponential Riordan array given by F = 3 m m m 6m (m 3) ( m 6m + 33 ) 0 0(m 3) 0 [ ] dn(z,m) cn(z,m), z is 3 The non-zero elements of the first column of matrix F are associated to the 34

138 matrix: G = The row sums of G form the sequence (,, 8, 3, 8, 5, ) which corresponds to A0894 having egf e z cosh (z) 54 The submatrix of [ ] dn(z,m), z cn(z,m) 4 The coefficient matrix of the proper exponential Riordan array given by H = 4 m (m 4) m 36m (m 4) ( m 36m + 56 ) 0 0(m 4) 0 [ ] dn(z,m) cn(z,m), z is 4 The non-zero elements of the first column of matrix H are associated to the matrix: I = The row sums of I form the sequence (, 3,, 83, 64, 4763, ) which corresponds to A having the egf cosh 3 (z) 55 The submatrix of [ ] dn(z,m), z cn(z,m) 5 The coefficient matrix of the proper exponential Riordan array given by J = 5 m (m 5) m 46m (m 5) ( m 46m + 85 ) 0 0(m 5) 0 [ ] dn(z,m) cn(z,m), z is 5 35

139 The non-zero elements of the first column of matrix J is associated to the matrix K = The row sums of K form the sequence (, 4, 40, 544, 830, 3584, ) which corresponds to A098 having the egf cosh 4 (z) 56 The submatrix of [ ] dn(z,m), z cn(z,m) 6 The coefficient matrix of the proper exponential Riordan array given by L = 6 m (m 6) m 56m (m 6) ( m 56m + 0 ) 0 0(m 6) 0 [ ] dn(z,m) cn(z,m), z is 6 The non-zero elements of the first column of the matrix M is associated to the matrix: M = The row sums of M form the sequence (, 5, 65, 05, 6465, 68805, ) which corresponds to A8 having the egf cosh 5 (z) 57 The submatrix of [cn(z, m), sn(z, m)] The coefficient matrix of [ cn(z, m), sn(z, m) ] is given by L = m (m + ) 0 4(m + 4) m + 74m (m + 3) 0 36

140 The non-zero elements of the second column of L are associated to the matrix M = The matrix M can be transformed by multiplying with ( ) n+ where n is the row number st n =,, 3, to obtain the matrix N = The elements of the first column of N forms the sequence (, 7, 6, 547, 49, 4487, ) which corresponds to A having egf 3e 9z + ez 4 The row sums of N form the sequence (, 8, 36, 3968, 76896, 848, , ) corresponding to the non zero elements of A0483 having egf tan(z) 58 The submatrix of The coefficient matrix of [, ] sn(z,m) +sn(z,m) [ ] sn(z,m), +sn(z,m) is given by P = m (m + 7) m + 74m (m + 7) 0 The second column of P is generated from the expansion of sn(z,m) +sn(z,m) non-zero elements of the second column of P are associated with the matrix Q = The The matrix Q can be transformed by multiplying with ( ) n+ where n is the 37

141 row number st n =,, 3, to obtain the matrix R = The row sums of R form the sequence (, 8, 56, 7408, 0366, , , ) which corresponds to A5365 known as the generalized Riemann zeta function ( ) n (( 6n+3 n ) ζ( n ) ζ( n ) ) The sequence A5365 can alternatively be generated from (n!)[z n ] sec(z), or in other words, sech(z) is the gf of the aerated sequence 53 Embedded Riordan arrays from Riordan arrays generated by Elliptic Functions In this section [ we apply] the technique of forming a new Riordan array using the formula d(z), h (z) z from the original Riordan array [d(z), h(z)], briefly introduced in (5) 53 The embedded Riordan array of [cn(z, m), sn(z, m)] The Riordan array [cn(z, ] m), sn(z, m)] produces the embedded Riordan array A = [cn(z, m), sn(z,m) z The coefficient matrix corresponding to A is given by B = m m + 0 (4m + 7) 0 The matrix B is associated to the production matrix C = (m + ) 0 (m ) 0 6m

142 53 The embedded Riordan array of [dn(z, m), sn(z, m)] The Riordan array [dn(z, ] m), sn(z, m)] produces the embedded Riordan array A = [dn(z, m), sn(z,m) z The coefficient matrix corresponding to A is given by B = m m 0 0 m(m + 4) 0 (7m + 4) 0 The matrix B is associated to the production matrix C = m m 0 ( m)m 0 9m The embedded Riordan array of [cn(z, m), zcn(z, m)] The Riordan array [cn(z,] m), zcn(z, m)] produces the embedded Riordan array A = [cn(z, m), (zcn(z,m)) z [ cn(z, m), zcn(z, m) ] The coefficient matrix corresponding to A is given by B = m The matrix B is associated to the production matrix C = m The embedded Riordan array of [dn(z, m), zdn(z, m)] The Riordan array A = [dn(z, m), ] zdn(z, m)] produces the embedded Riordan array B = [dn(z, m), (zdn(z,m)) z [ dn(z, m), zdn(z, m) ] The coefficient matrix corresponding to A is given by B = m m 0 0 m + 4m 0 30m 0 39

143 The matrix B is associated to the production matrix C = m m 0 4( m)m 0 m 0 54 Riordan arrays derived from the r-shifted central triangles of a given Riordan array The family of r-shifted central coefficients a n+r,n+r of a Riordan array were originally introduced by the authors Zheng and Yang [6] A family of matrices c(a; r) known as the r-shifted central triangles of the Riordan array A having general term a n+r,n+k+r can be constructed from the original Riordan array A These new matrices arising from A have the r-shifted central coefficients as their leftmost column with all s on their principal diagonal The outcome of trying to characterize the family of matrices c(a; r) results in a one parameter family of Riordan arrays These results for c(a; r) when r = are summarized below Theorem 54 The shifted central triangle c(a; ) of the Riordan array A = [g(z), zf(z)] is a Riordan array which admits the following factorization [ ] φ c(a; ) = f(φ), φ A, where ( ) z φ(z) = Rev and φ (z) = d f(z) dz φ(z) Corollary 54 We have Corollary 543 We have c(a; ) = c(a; ) = [ φ g(φ) ] f(φ), φf(φ) [ ] f(z) z φ ( z f(z) ), [g(z), zf(z)] f(z) 40

144 Corollary 544 c(a; ) = f( v) v ( ),, g( v) φ v f( v) f( v) where v(z) = zf(z) Corollary 545 Let A = [f(z), zf(z)] be a member of the Bell subgroup of the Riordan group Then c(a; ) = [φ, φf(φ)], and c(a; ) = ( φ v f( v) v ), f( v) We investigate some examples corresponding to each of these results below 54 [ ] z cn(z, m), sn(z,m) We verify the results of Corollary (543) for the case of the the Riordan array A = [ z ] cn(z, m), [ cn(z, m), z ns(z, m) ] sn(z, m) The Riordan array A produces the Riordan array B = by applying the formula [ z ( msn(z m) ) cd ( sn (sn(z m) m) m ) f(z) = [ f(z) φ ( z ), f(z) sn(z m) The coefficient matrix of B is given by 8 5 ] z f(z) where z sn(z, m) and φ(z) = sn (z, m), sn(z m) (m + ) (m + ) ((m 6)m + ) 0 8(m + ) (m(m + 4) + ) (m + ) 0 ] 4

145 55 A one-parameter family of Riordan arrays derived from a given Riordan array For every integer s a transformation of a Riordan array can be defined which yields another Riordan array By setting a value for the integer s the effects of these transformations on some Jacobi Riordan arrays are evaluated for new combinatorial results arising from the new Riordan matrices 55 Examples using the inverse transformation T (s) 55 The case of [g, f] = [cd(z, m), z] We apply the formula T (s) = [g, f] [ ψ (z) ( ) ] s ψ z, ψ where ψ(z) = z +f(z), to the original Riordan array [g, f] = [cd(z, m), z] In particular, if s = we have that ψ = z + z = [ ψ (z) [ T () = ( ) s [ ψ, ψ] = z ( + z) 3 cd(z, m), z + z ( + z) 3, z + z ] ] The matrix representing T () is given by T = 3 m m m (m 78)m (m ) 6(m 3) 4 0 5((m 46)m + 3) 5((m 6)m + 965) 50(m 5) 0(m 43) 35 The production matrix of T is given by T p = m 5 0 m (m 5) 7 (m )(m + 4) 6 6m 3(m 6) 9 4

146 Remark : If m = The coefficient matrix of T becomes [ The coefficient matrix T = T = (+z), 3 ] [ z +z ψ (z) ( ) ] s ψ z, ψ If we multiply T by ( ) n+ where n represents the row number of T starting from, we get the matrix T = The matrix T corresponds to A0639 which is the coefficient triangle of the generalized Laguerre polynomials n! L(n,, z) for rising powers of z The row sums of the signed triangle T are (,, 5, 4, 37, 34, 887, ) corresponding to e z z A0697 with egf ( z) 3 Remark : transformation T (s) by The exponential Riordan array [ ] z (+z), 3 +z arising from the has a tri-diagonal production matrix for m = given B = [ This indicates that the inverse matrix of (+z), 3 ] z +z is associated to the coefficients of a family of orthogonal polynomial sequences Remark 3: The r and c generating function of T are respectively r(z) = z c(z) = 3 3z 43

147 55 The case of [g, f] = [cd(z, m), sn(z, m)] We apply the formula T (s) = [g, f] [ ψ (z) ( ψ z ) s, ψ ] where ψ(z) = z +f(z), to the original Riordan array [g, f] = [cd(z, m), sn(z, m)] In particular, if s = we have that ψ = z + sn(z, m) = [ ψ (z) ( ) s ψ, ψ] = z [ ] zcn(z m)dn(z m) + sn(z m) + z (sn(z m) + ) 3, + sn(z, m) [( mz ) cd ( sn (z m) ) ] m [g, f] = z, sn (z, m) T () equals to [ cd ( sn ( sn (z m) m ) m ( z( cn ( sn (z m) m ) )dn ( sn (z m) m ) + z + ) (z + ) 3, ] z + sn(z, m) 44

148 Chapter 6 Riordan arrays and Solutions to Differential Equations 6 The Sturm-Liouville equation The Sturm-Liouville equation [8] is a real second order differential equation of the form [ d p(x) dy ] + q(x)y = λw(x)y (6) dx dx The equation (6) is derived under general conditions by rewriting a given second order homogenous differential equation in one dimension of the form a(x) d d y(x) + b(x) y(x) + c(x)y(x) = 0, dx dx in another form of differential equation involving a self-adjoint under suitable boundary conditions [86] In this sense, the Sturm-Liouville operator can be defined by Ly = 0 where L = d dx p(x) d dx q(x) 45

149 The eigen-value equation associated with L can be written as Ly + λwy = 0 d [ p(x) dy(x) ] q(x)y(x) + λw(x)y(x) = 0, dx dx where λ is the eigen value corresponding to the eigen function y(x) satisfying the boundary conditions, the real valued function ω(x) > 0 is the weight function A catalogue providing details of over 50 examples of Sturm-Liouville differential equations has been presented by Everitt [38] Most of these differential equations are directly related to problems in mathematical physics Riordan arrays can represent the solution to systems of differential equations The commonest examples come from Sturm Liouville systems with parameter λ that depends on n In the account that follows a connection between the solutions to some Sturm-Liouville differential equations with ordinary and exponential Riordan arrays will be outlined For the purpose of this investigation the case of the Laguerre equation, Bessel equation, Hermite equation, Morgan-Voyce equation and Chebyshev equation are considered Each of these equations will be treated in such a way as to show that the coefficient arrays corresponding to the polynomial sequences that make up their solutions are Riordan arrays 6 Laguerre Polynomials The general form of the Laguerre differential equation [] is xy + ( x)y + λy = 0 (6) In addition, (6) is considered a special case of the general associated or generalized Laguerre differential equation defined by xy + (v + x)y + λy = 0 (63) where λ and v are real numbers The solutions to the Laguerre differential equation (6) are given by a polynomial sequence known as Laguerre polynomials which are most often denoted as L 0, L, L, and can be expressed 46

150 explicitly by the formula L n (x) = n! n ( ) k ( ) n x k k! k k=0 The generating function of the Laguerre polynomials is given by xt e ( t) = t n=0 L n (x) n! The Laguerre polynomials satisfy the 3-term recurrence relation (n + )L n+ (x) = (n + x)l n (x) nl n (x) (64) [ ] [ ] It can be verified that L n (x) = t, t t e xt t, t t e xt The general formula for the polynomial sequence L n (x) is computed using the FTRA (5) and the method of coefficients as follows: [ ] [t n ] t, t e xt = n! [t n xt ] e t t t = n! [t n ( ) k x k t k ] t k!( t) k k=0 = n! [t n ( ) k x k t k ] ( t) (k+) k! k=0 = n! [t n ( ) k x k t k ( ) (k + ) ] ( t) j k! j k=0 j=0 = n! [t n ( ) k x k t k ( ) k + ] ( ) j t j k! j k=0 j=0 = n! [t n ( ) k x k t k ( ) k ] ( ) j ( ) j t j k! j k=0 j=0 = [ t n k] n n!( ) k x k ( ) k t n k k! n k k=0 n n!( ) k ( ) n = x k k! k k=0 = (L n (x)) n N 47

151 The coefficient matrix of the Laguerre polynomials L n (x) is given by A = The production matrix associated to A is given by P A = Remark: The production matrix P A is tridiagonal which indicates that A forms a family of orthogonal polynomial sequences We verify that the polynomial sequence L n (x) forms an orthogonal polynomial sequence satisfying the 3-term recurrence relation (64) We proceed by calculating the inverse of the matrix A corresponding to the coefficient matrix L n (x) and the production matrix of A to determine its recurrence coefficients The coefficient matrix A is given by A = The production matrix of A is given by P A = Remark: The tri-diagonal nature of the production matrix P A of A shows that the matrix A is the coefficient matrix of a family of orthogonal polynomial sequences Recall from section (7) that for a polynomial sequence (p n (x)) n N satisfies 48

152 the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x) for n From the production matrix P A we get α n = n + and β n = n with the initial conditions L 0 (x) = and L (x) = x Thus, L n+ (x) = (n + x)l n (x) n L n (x) for n 6 Hermite Polynomials The Hermite differential equation is a second order differential equation given by d y dy x + λy = 0 (65) dx dx The solutions to the Hermite equation (65) are called the Hermite polynomials There are two forms of the Hermite polynomials These are the probabilist Hermite polynomials and the physicist Hermite polynomials denoted He n annd H n respectively The relationship between the two types of Hermite polynomials is such that He n (x) = n Hn ( x) 6 The Probabilist Hermite polynomials He n The generating function of the probabilist Hermite polynomials is given by e xt t = n=0 He n (x) tn n! The probabilists Hermite polynomial He n (x) can be defined as n ( ) k x n k He n (x) = n! k!(n k)! k k=0 n k=0 n! + ( )n k ( ) n k k!( n k )! x k The recursion relation of the Hermite polynnomials He n (x) is given by He n+ (x) = xhe n (x) nhe n (x) (66) 49

153 The coefficients of the] Hermite polynomials He n can be represented by the Riordan array [e t, t such that [e t, t ] e xt = (He n (x)) n N The general formula for the polynomial sequence He n (x) is computed using the FTRA (5) and the method of coefficients as follows ] [t n ] [e t, t e xt = n![t n ]e t +xt = n![t n ( t ] + xt)k k! k=0 = n! ( ) k t k! [tn ] t k + x = n! k! [tn ] = n! k! [tn ] = n! k! [tn ] = = = = = k=0 t (t k k=0 n! k k! [tn ] k=0 ( + x t )) k ( t k + x ) k t k=0 ( t k ( x )) k t k=0 ( t k ( ) k x t ) k n! k k! [tn ] k ( ) ( ) j k x t k ( ) k j t k j=0 n! k k! [tn ] k ( ) ( ) j k x t k ( ) k ( ) j j t k j=0 n! k k! [tn ] ( ) k t ( ) k k j ( ) j (x) j j k j n! ( ) k k ( ) k ( ) k n (x) k n k! k n k=0 50

154 The coefficient matrix of the Hermite polynomials He n (x) is given by A = The production matrix associated to A is given by P A = Remark:The production matrix P A is tridiagonal which indicates that A forms a family of orthogonal polynomial sequences We verify that the polynomial sequence He n (x) forms an orthogonal polynomial sequence satisfying the 3-term recurrence relation (66) We proceed by calculating the inverse of the matrix A corresponding to the coefficient matrix He n (x) and the production matrix of A to determine its recurrence coefficients The coefficient matrix A is given by A = The production matrix of A is given by P A =

155 Remark: The tri-diagonal nature of the production matrix P A of A shows that the matrix A is the coefficient matrix of a family of orthogonal polynomial sequences defined by the Hermite polynomials He n (x) Recall from section (7) that for an orthogonal polynomial sequence (p n (x)) n N satisfies the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x), n From the production matrix P A we get α n = 0 and β n = n Thus, He n+ (x) = xhe n (x) nhe n (x) for n with the initial conditions He 0 = and He = x 6 The Physicist Hermite polynomials H n (x) The physicists Hermite polynomials H n (x) may be defined as n H n (x) = n! k=0 The generating function for H n (x) is given by e xt t = ( ) k (x) n k k!(n k)! n=0 H n (x) tn n! The recursion relation of the Hermite polynomials H n (x) is given by H n+ (x) = xh n (x) nh n (x) (67) The coefficients[ of the ] Hermite polynomials H n (x) can be represented by the Riordan array e t, t such that [ ] e t, t e xt = (H n (x)) n N 5

156 The general formula for the polynomial sequence H n (x) is computed using the FTRA (5) and the method of coefficients as follows: [ ] [t n ] e t, t e xt = n![t n ]e t +xt = n![t n ] = n! k! [tn ] = n! k! [tn ] = n! k! [tn ] = n! k! [tn ] = n! k! [tn ] ( t + xt) k k=0 k! t k ( t + x) k k=0 ( t (t k + x )) k t ( t k + x ) k t ( t ( k x )) k t ( t k ( ) k x ) k t k=0 k=0 k=0 k=0 = n! k! [tn ] k = n! k! [tn ] k = n! k! [tn ] k = n! k! [tn ] k = n! k! t k ( ) k t k ( ) k t k ( ) k k j=0 k j=0 ( ) ( k x j t ) j ( ) ( k x ( ) j j t k ( k j j=0 ) ( ) j ( x t ) j ) j ( ) k t ( ) k k j ( ) j (x) j j j ( ) k ( ) k ( ) k n (x) k n k n k=0 The coefficient matrix of the Hermite polynomials H n (x) is given by A =

157 The production matrix associated to A is given by P A = Remark: The production matrix P A is tridiagonal which indicates that A forms a family of orthogonal polynomial sequences We verify that the polynomial sequence H n (x) forms an orthogonal polynomial sequence satisfying the 3-term recurrence relation (67) We proceed by calculating the inverse of the matrix A corresponding to the coefficient matrix H n (x) and the production matrix of A to determine its recurrence coefficients The coefficient matrix A is given by A = The production matrix of A is given by P A = Remark: The tri-diagonal nature of the production matrix P A of A shows that the matrix A is the coefficient matrix of a family of orthogonal polynomial sequences defined by the Hermite polynomials H n (x) Recall from section (7) that for an orthogonal polynomial sequence (p n (x)) n N satisfies the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x) From the production matrix P A we get α n = 0 and β n = n (by applying a 54

158 scaling factor of to P A gives H n+ (x) = xh n (x) nh n (x) as defined in (67) 63 The Reverse Bessel Polynomials The reverse Bessel polynomial satisfies the differential equation given by xθ n(x) (x + n)θ n(x) + nθ n (x) = 0 (68) The explicit formula of the reverse Bessel polynomial is given by where k = 0,,, n θ n (x) = n k=0 (n k)! n k k!(n k)! xk (69) The 3 term recurrence relation of the reverse Bessel polynomial is given by θ n (x) = (n )θ n (x) + x θ n (x) The [ proper exponential Riordan array of the reverse Bessel polynomial is R BS = t, ] t represents the solution of (68), in the sense that R BS {, x!, x!, x3 3!, }T = {θ 0 (x), θ (x), θ (x), } T The coefficient matrix of the reverse Bessel polynomial is given by R BS = The production matrix of R BS is given by

159 The general formula for the entries of the coefficient matrix of the Reverse Bessel polynomials are calculated using its Riordan array representation as follows: [, ] x x = n! ( ) k k! [xn ] x x = n! ( ) k k! [xn ] ( ) j ( x) j/ ( x) / j = n! k ( ) k k! [xn ] ( ) j ( x) j j = n! k! [xn ] = n! k! k j=0 j=0 k j=0 ( k j = n! k! ( )n = n! k! ( )n ( ) k ( ) j j i )( ) j ( j n k j=0 k j=0 ( k j ( k j )( j n ( j ) ( ) i x i i ) ( ) n ) ( ) j )( ) j ( j n The Mathematica code below computes the coefficient matrix of the reverse Bessel polynomials using the general formula Table [ Table [ n! k! ( )n Sum [ Binomial[k, j]binomial [ j, n] ( ) j, {j, 0, k} ], {k, 0, 0} ], {n, 0, 0}] ) 64 Chebyshev Polynomials The Chebyshev differential equation given by ( x ) d y dx x dy dx + λ y = 0 (60) In order to determine that the Chebyshev equation satisfies the Sturm-Liouville equation (6) it is divided by x to get the equation ( x )y x λ x y + y = 0 (6) x 56

160 such that ( [ x )y x x y corresponds to d of the Sturm-Liouville form (6) with dx p(x) = x, q(x) = 0, w(x) = ( x ) dy dx x ] in the context The solutions to the Chebyshev equation are known as the Chebyshev polynomials The Chebyshev polynomials are of two types referred to as the Chebyshev polynomials of the first kind and the Chebyshev polynomials of the second kind 64 The Chebyshev polynomials of the second kind The generating function of the Chebyshev polynomial of the second kind[] is u(t, x) = xt + t = U n (x)t n (6) The formula for the general term of the chebyschev polynomial of the second kind is given by k=0 ( The ordinary Riordan array R c = differential system in the sense that n=0 n ( ) n k U n (x) = ( ) k x n k k ) t +t, +t R c {, x, x, } T = {U 0 (x), U (x), U (x), } T represents the solution of the 64 The Scaled Chebyshev Polynomials The family of scaled Chebyshev polynomials n ( ) n k y n (x) = U n (x/) = ( ) k x n k k k=0 satisfies the system of differential equations (4 x )y n 3xy n + n(n + )y n = 0 57

161 The Riordan array R = represents the solution, in the sense that ( ) + t, t + t R {, x, x, } T = {y 0 (x), y (x), y (x), } T The general formula for the polynomial sequence y n (x) is computed using the FTRA (5) and the method of coefficients as follows: ( ) [t n ] + t, t + t xt +t = [t n ] ( ) t x +t = [t n ] ( + t t x = [t n ( xt ] + t + t = [t n ] + t = [t n ] k=0 = [t n k ] = [t n k ] = [t n k ] = n k=0 k=0 k=0 t k +t ) ) k t k ( + t ) k xk ( + t xk ) k+ ( ) (k + ) t j x k j ( ) k + + j ( ) j t j x k j ( ) k + j ( ) j t j x k j j=0 j=0 j=0 ( n+k k = (y n (x)) n N ) ( ) n k ( + ( ) n k ) x k Remark: From the above results we have j = n k = j = n k so that ( k + n k ) ( ) n k ( + ( ) n k ) n k 58

162 and also n + k n k = k 65 Morgan-Voyce Polynomials The Morgan-Voyce polynomials are can be explicitly defined in two types denoted by B n (x) and b n (x) 65 The Morgan-Voyce Polynomials B n (x) The Morgan-Voyce polynomial B n (x) is given explicitly by the formula B n (x) = n ( ) n + k + x k (63) n k k=0 and it satisfies the ordinary differential equation x(x + 4)y + 3(x + )y n(n + )y = 0 (64) The Morgan-Voyce polynomials B n (x) can also be defined by the 3-term recurrence relation with B 0 (x) = and B (x) = + x B n (x) = (x + )B n (x) B n (x) (65) ( The proper ordinary Riordan array R MB = ) t ( t) represents the so- lution of (64), in the sense that ( t), R MB {, x, x, } T = {B 0 (x), B (x), B (x), } T The general formula for the polynomial sequence B n (x) is computed using the 59

163 FTRA (5) and the method of coefficients as follows: ( [t n ] ( t), t ( t) ) xt ( t) ) t ( t) = [t n ] ( x = [t n ] ( ( t) x = [t n ] ( t) = [t n ] ( t) = [t n ] k=0 k=0 ) t ( t) ) k ( xt ( t) x k t k ( t) k x k t k ( t) (k+) k=0 ( ) k + + j = [t n k ] ( ) j ( t) j x k j j=0 n ( ) k + n k + = [t n k ] t n k x k n k k=0 n ( ) n + k + = x k n k k=0 = (B n (x)) n N The coefficient matrix of the Morgan-Voyce polynomials B n (x) is given by A = The production matrix associated to A is given by P A = Remark: The production matrix P A corresponds to the structural pattern of the production matrices for proper ordinary Riordan arrays defined in sec- 60

164 tion (6) We verify that the polynomial sequence B n (x) forms an orthogonal polynomial sequence satisfying the 3-term recurrence relation (65) We proceed by calculating the inverse of the matrix A corresponding to the coefficient matrix B n (x) and the production matrix of A to determine its recurrence coefficients The coefficient matrix A is given by A = The production matrix of A is given by P A = Remark: The tri-diagonal nature of the production matrix P A of A shows that the matrix A is the coefficient matrix of a family of orthogonal polynomial sequences defined by the Morgan-Voyce polynomials B n (x) Recall from section (7) that for an orthogonal polynomial sequence (p n (x)) n N satisfies the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x) From the production matrix P A we get α n = and β n = Thus, B n (x) = (x + )B n (x) B n (x) with B 0 (x) = and B (x) = + x as defined in (65) 6

165 65 The Morgan-Voyce Polynomials b n (x) The Morgan-Voyce polynomials b n (x) is given explicitly by and it satisfies the equation b n (x) = n k=0 ( ) n + k x k (66) n k x(x + 4)y + (x + )y n(n + )y = 0 (67) The Morgan-Voyce polynomials b n (x) can also be defined by the 3-term recurrence relation b n (x) = (x + )b n (x) b n (x) (68) with ( b 0 (x) = ) and b (x) = + x The proper ordinary Riordan array R Mb = t, t ( t) represents the solution of (67), in the sense that R Mb {, x, x, } T = {b 0 (x), b (x), b (x), } T The general formula for the polynomial sequence b n (x) is computed using 6

166 the FTRA (5) and the method of coefficients as follows: ( ) [t n ] t, t ( t) xt t = [t n ] ( ) t x ( t) = [t n ] ( ) t t x ( t) = [t n ( ) k xt ] t ( t) k=0 = [t n ] x k t k t ( t) k k=0 = [t n ] x k t k ( t) (k+) k=0 ( ) k + + j = [t n k ] ( ) j ( t) j x k j j=0 n ( ) k + n k = [t n k ] t n k x k n k k=0 n ( ) n + k = x k n k k=0 = (b n (x)) n N The coefficient matrix of the Morgan-Voyce polynomials b n (x) is given by A = The production matrix associated to A is given by P A = Remark: The production matrix P A corresponds to the structural pattern of the production matrices for proper ordinary Riordan arrays defined in sec- 63

167 tion (6) We verify that the polynomial sequence b n (x) forms an orthogonal polynomial sequence satisfying the 3-term recurrence relation (68) We proceed by calculating the inverse of the matrix A corresponding to the coefficient matrix b n (x) and the production matrix of A to determine its recurrence coefficients The coefficient matrix A is given by A = The production matrix of A is given by P A = Remark: The tri-diagonal nature of the production matrix P A of A shows that the matrix A is the coefficient matrix of a family of orthogonal polynomial sequences defined by the Morgan-Voyce polynomials b n (x) Recall from section (7) that for an orthogonal polynomial sequence (p n (x)) n N satisfies the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x) From the production matrix P A we get α n = and β n = Thus, b n (x) = (x + )b n (x) b n (x) with b 0 (x) = and b (x) = + x as defined in (68) 64

168 6 Boubaker polynomials and the solutions of some Differential Equations The Boubaker polynomials form a monic orthogonal sequence of polynomials which originated from an attempt to determine the discrete form of solution to a non-linear heat transfer problem related to the spray pyrolysis disposal performance [6] This led to the development of the Boubaker polynomials expansion scheme denoted BPES, which is a resolution protocol with several applications in applied physics, engineering and mathematical problems Some collection of the applications of BPES have been listed and summarized in the 3 page paper [7] The main advantage of the BPES is that it requires that the boundary conditions are satisfied irrespective of the main features of the equation The BPES is based on the Boubaker polynomials The recursive definition of the Boubaker polynomials is given by B n (x) = xb n (x)-b n (x) if n = 0 x if n = x + if n = otherwise, n Z The ordinary generating function of the Boubaker polynomials is given by B(x, t) = + 3t + t(t x) The characteristic differential equation of the Boubaker polymials is given by A n y + B n y C n y = 0 where A n = (x )(3nx + n ) B n = 3x(nx + 3n ) C n = n(3n x + n 6n + 8) 65

169 The explicit closed form formula of the Boubaker polynomials is given by B n (x) = n/ k=0 n 4k n k ( n k K ) ( ) k x n k The first 5 Boubaker polynomials are B 0 (x) = B (x) = x 0x 0 B (x) = x 0x + x 0 B 3 (x) = x 3 0x 0 + x + 0 B 4 (x) = x 4 0x 3 + 0x + 0x B 5 (x) = x 5 + 0x 4 x 3 + 0x 3x + 0x 0 The relationship between the Boubaker polynomials and the Chebyshev polynomials [9] is given by B n (x) = 4x n d dx T n(x) T n (x) B n (x) = T n (x) + 4xU n (x) where T n denotes the Chebyshev polynomials of the first kind and U n denotes the Chebyshev polynomials of the second kind The Riordan array representation of the coefficients of the Boubaker polynomials is given by B = The coefficient matrix of B is given by ( ) + 3x + x, x + x B =

170 The production matrix of B is given by P B = The inverse of the matrix B is given by B = The production matrix of B is given by P B = The tri-diagonal nature of the production matrix P B of B shows that the matrix B is the coefficient matrix of a family of orthogonal polynomial sequences defined by the Boubaker polynomials B n (x) Recall from section (7) that for an orthogonal polynomial sequence (p n (x)) n N satisfies the 3-term recurrence relation p n+ (x) = (x α n )p n (x) β n p n (x) From the production matrix P B we get α n = 0 and β n = Thus, we have B n (x) = xb n (x) B n (x) where the initial conditions B 0 =, B (x) = x, B (x) = x + are satisfied The 4 key properties derived from the Boubaker polynomials pertains to its polynomial non-zero orders being divisible by 4 This forms the basis behind its already existing and ongoing research to analytic solutions of the differential equations that it solves These properties when considered for n = 4q, q = N, N Z >0 are listed as: 67

171 3 4 B n (x) x=0 = N 0 n>0 B n (α i ) = 0, α i a non-zero positive root of B n n>0 n>0 n>0 db n(x) dx = 0 x=0 and H n = B 4k (r n) = db n (x) dx = 8 3 N(N ) x=0 B 4q (rn) q= B 4(n+) (r n) 4rn[ r n] n + 4r 3 n Some of the differential equations having analytic solutions arising from the implementation of the BPES technique are listed below The Lane-Emden equations of the first and second kind respectively governing polytropic and isothermal gas spheres, are given in Boubaker & Gorder[8] as follows y + x y + y n = 0 with initial conditions y(0) =, y (0) = 0 y + x y = e y with initial conditions y(0) = 0, y (0) = 0 The Bloch NMR Flow equation given in Awojoyogbe et al[4] as follows x d M y dx satisfying the initial conditions + α dm y dx + βxm n y = 0 α=,β=;n=,3,4,5 M y (0) = ; dm y(0) dx where β is a unique constant associated to the NMR system under investigation A differential equation from a nonlinear circuit investigated in Vazquez et al[3] is given by dq(t) dt = 0 + α R q3 (t) = I, q(0) = 0 Other examples of differential equations solved using the BPES technique are found in [59,, ] 68

172 Chapter 7 Elliptic Functions and the Travelling Wave Solutions to the KdV equation 7 Introduction The presence of solitary water waves moving for a considerable distance down a narrow channel was first observed and reported by John Scott Russell, a Scottish naval engineer back in 834 This was followed up by Korteweg and de Vries(895) mathematical formulation of a third order nonlinear PDE referred to as the KdV equation [60] The role of this KdV equation which admits traveling wave solutions was to determine an approximate theory to model the evolution of the propagation of nonlinear shallow water waves There are many different variants of the KdV equation The standard form of the KdV equation is given by u t 6uu x + u xxx = 0 (7) Sometimes it is +6 used in the above equation In the KdV equation, the variable t denotes the time, the variable x denotes the space coordinate along the canal and the function to be determined denoted u = u(x, t) represents the elevation of the fluid above the bottom of the canal [3] In 965 Kruskal and Zabusky showed that the KdV equation admits analytic 69

173 solutions known as Solitons [3] Solitons can be described as propagating pulses or solitary waves which are stable by maintaining their shape and do not disperse with time The KdV equation can also have more than one soliton solution which can move towards each other, interact and then emerge at the same speed with no change in shape but only a time lag [93] There are several methods available for solving different forms of non-linear wave equations in mathematical physics [5] The method applied in determining the explicit exact solutions to the KdV equation provides a good entry point on how to approach the solutions of other such non-linear wave equations In particular, using appropriate elliptic functions to determine the explicit solutions for the KdV equation is important as it leads to periodic wave solutions and not only solitary wave solutions 7 Elliptic functions as solutions to the KdV equation In this section we shall consider the two types of elliptic functions that solve the KdV equation These are the cnoidal travel wave solution determined by elliptic cn and the elliptic Weierstrass Each of these solutions characterized by their doubly periodic nature are elaborated below 7 Elliptic cn as a solution to the KdV equation In this subsection we verify the solution to the KdV equation (7) with the aid of the symbolic computational algebra system Mathematica, by using the cnoidal traveling wave solution of the form u(ξ) = a + bcn (ξ, m) proposed but unsolved in [36] Consider the nonlinear PDE of the form H(u, u t, u x, u xx, ) = 0 (7) for which equation (7) satisfies H is expressible as a polynomial in u where u = u(x, t) is the unknown function such that u is the physical field and x, t represents its independent variables The simplest mathematical wave function 70

174 is of the form u(x, t) = f(x ct) which can easily be shown to be a solution to the PDE u t + cu x = 0 In order to obtain an ODE we transform using the simplest traveling wave equation u(x, t) = u(ξ), ξ = x ct (73) u(x, t) = u(ξ), ξ = k(x ct) where c is the wave speed moving in either the left or right direction, k is the wave number Using (73) the PDE (7) can be transformed to a nonlinear ordinary differential equation given by F (u, u, u, u, ) F where the prime is such that := d dξ ( u, du ) dξ, d u d ξ, d3 u d 3 ξ, and F is expressible as a polynomial in terms of u(ξ) As a result we have that substituting u(ξ) into (7) we get the ODE c du dξ 6udu dξ + d3 u = 0 (74) dξ3 Integrating equation (74), we get the second order differential equation cu 3u + d u dξ = k (75) where k is the constant of integration Given that k vanishes if x and u 0, we have cu 3u + d u = 0 (76) dξ The steps to determine the exact solution of (7) are outlined below Using u(ξ) = a + bcn (ξ, m) we have d u dξ = b ( cn(ξ m) dn(ξ m) + mcn(ξ m) sn(ξ m) + dn(ξ m) sn(ξ m) ) Substituting u(ξ) = a + bcn (ξ, m) and (77) into (76) we get (77) 7

175 c ( a + bcn(ξ m) ) 3 ( a + bcn(ξ m) ) + b ( cn(ξ m) dn(ξ m) + mcn(ξ m) sn(ξ m) + dn(ξ m) sn(ξ m) ) (78) Simplifying and expanding equation (78) gives 3a 6abcn(ξ m) ac 3b cn(ξ m) 4 bccn(ξ m) 4bcn(ξ m) dn(ξ m) + bcn(ξ m) + bdn(ξ m) sn(ξ m) (79) Recall that cn ξ = sn ξ and dn ξ = m sn ξ (70) where m is the modulus with 0 m Substituting (70) for sn and dn in terms of cn in (79) results to 3a 6abcn(ξ m) ac 3b cn(ξ m) 4 bccn(ξ m) 4b ( m ( cn(ξ m) )) cn(ξ m) + b ( cn(ξ m) ) ( m ( cn(ξ m) )) + bcn(ξ m) (7) Expanding (7) gives 3a 6abcn(ξ m) ac 3b cn(ξ m) 4 bccn(ξ m) 6bm cn(ξ m) 4 + 8bm cn(ξ m) 4bcn(ξ m) bm + b (7) Extracting the coefficients of (7) from cn i where i = 0,,, 3, 4 we get { 3a ac bm + b, 0, 6ab bc + 8bm 4b, 0, 3b 6bm } (73) Setting the non-zero elements in (73) to zero and solving the system of equations for a, b, c in terms of m, the following set of solutions are obtained : 7

176 Solution - If < m < then a = 6 6m4 6m m4 64m + 6, b = m, c = 6m 4 6m + 6 Solution - If m > m < then a = 6 64m4 64m m4 6m + 6, b = m, c = 6m 4 6m Solution 3- If m > m < then a = 6 6m4 6m m4 64m + 6, b = m, c = 6m 4 6m Substituting the values of a, b, c obtained in the step above into u(ξ) = a + bcn (ξ, m) where ξ = x ct, we get the following solutions outlined below for (7) in the form u(x, t) as required (a) For solution we get u(x, t) = m cn ( 6m 4 6m + 6t x m) 6 6m4 6m m4 64m + 6 (b) For solution we get u(x, t) = m cn ( 6m 4 6m + 6t x m) 6 6m4 6m m4 64m + 6 (c) For solution 3 we get u(x, t) = m cn ( 6m 4 6m + 6t +x m) + 6 6m4 6m m4 64m + 6 In particular, we determine the solutions of (7) corresponding to m = 0 and the soliton solutions corresponding to m = Observing the conditions given in Solution and Solution 3 which are m > m < and m > m < respectively, these solutions corresponding to (b) and (c) above are u(x, t) = sech (4t x) and u(x, t) = 4 3 sech (4t + x) respectively The case for m = 0 in (a) resulted to u(x, t) = 4 3 solution was identified for this case and so no non-trivial The solutions of H in the form of u have physical significance The soliton solution u(x, t) = sech (4t x) in physical terms describes a trough of depth traveling to the right with speed 4 and not changing its shape On the other hand the soliton solution u(x, t) = 4 3 sech (4t+x) in physical terms describes a trough of depth traveling to the left with speed 4 and not changing its shape The plots corresponding to the traveling wave solutions of the KdV equation are depicted below 73

177 Figure 7: Plot depicting the KdV solution u(x, t) = sech (4t x) at t = 0 Figure 7: Plot depicting the KdV solution u(x, t) = 4 3 sech (4t + x) at t = 0 7 Elliptic Weierstrass as a solution to the KdV equation Another form of the KdV equation [5] equivalent to (7) is u t = 3 uu x + 4 u xxx Similar to (7) this equation describes the time evolution of the wave as it travels in one direction We look for a travelling wave solution to the KdV equation, of the form u(x, t) = w(x + ct) 74

178 We get cw = 3 ww + 4 w We integrate this equation to get cw = 3 4 w + 4 w + γ By multiplying the equation across by the integrating factor w we get cww = 3 4 w w + 4 w w + γ w A second integration gives Re-arranging the equation results to c w = 4 w3 + 8 (w ) + γ w + γ (w ) = w 3 + 4cw 8γ w 8γ Now for any constant ω, a solution to this equation is given by w(z) = (z + ω, {g, g 3 }) + 3 c, where Thus, g = 4 3 (c 3γ ), and g = 8c3 7 4cγ γ 3 u(x, t) = (x + ct + ω, g, g 3 ) + c 3 is a solution to the KdV equation for any choice of ω, g, g 3 and c A non-elliptic function solution of the KdV is of the form u(x, t) = 8k (e kx+k3 t + e kx k3 t ) This particular solution describes for any k, a solitary wave that travels at speed k and has a height of k 75

179 Figure 73: Plot depicting a Weierstrass solution for the KdV PDE 73 Riordan arrays determined from Elliptic cn 73 [ cn(ξ, m), cn(ξ, m) dξ ] The exponential Riordan array [ cn(ξ, m), cn(ξ, m) dξ ] belongs to the derivative subgroup (4) ( cn(ξ m) cn(ξ, m) + m dξ = ) E(am(ξ m) m) dn(ξ m) ξ msn(ξ m) m + ξ with its Taylor series expansion given by ξ ξ (m+)ξ ( m 3m ) m ξ +( m + 30m + ) ξ 9 +O ( ξ ) 835 The coefficient array of [ cn(ξ, m), cn(ξ, m) dξ ] is given by A = (m + ) m ( m + 3m + ) 0 56(3m + 8) The production matrix of A in terms of m is given by B = (m ) (m ) (m )(m ) 0 0(m )