A Study of Integer Sequences, Riordan Arrays, Pascal-like Arrays and Hankel Transforms

Transcription

1 UNIVERSITY COLLEGE CORK A Study of Integer Sequences, Riordan Arrays, Pascal-lie Arrays and Hanel Transforms by Paul Barry A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the College of Science, Engineering and Food Science Department of Mathematics Head of Department: Professor Martin Stynes Supervisor: Professor Patric Fitzpatric, Head of College of Science, Engineering and Food Science July 009

2 Contents Introduction Overview of this wor Preliminary Material 5 Integer sequences 5 The On-Line Encyclopedia of Integer Sequences 6 3 Polynomials 6 4 Orthogonal polynomials 7 5 Power Series 9 6 Ordinary generating functions 0 7 Exponential generating functions 8 Generalized generating functions 3 9 The Method of Coefficients 3 0 Lagrange inversion 5 Recurrence relations 6 Moment sequences 8 3 The Stieltjes transform of a measure 4 Orthogonal polynomials as moments 5 Lattice paths 6 Continued fractions 3 7 Hypergeometric functions 38 8 Transformations on integer sequences 43 9 The Hanel transform of integer sequences 45 0 Simple Pascal-lie triangles 46 3 Integer sequences and graphs 63 3 Notation 64 3 Circulant matrices The graph C 3 and Jacobsthal numbers The case of C The case of C The General Case of C r A wored example The case n 76 ii

3 39 Sequences associated to K r 77 4 Riordan arrays 8 4 The ordinary Riordan group 8 4 A note on the Appell subgroup The subgroup (g(x, xg(x The subgroup (, xg(x The exponential Riordan group A note on the exponential Appell subgroup Conditional Riordan arrays Generalized Riordan arrays 9 49 Egorychev arrays 9 40 Production arrays 9 5 The Deleham construction 5 5 Definition of the Deleham construction 5 5 The fundamental theorem 6 53 The Deleham construction and Riordan arrays 9 54 The Deleham construction and associahedra 30 6 Riordan arrays and a Catalan transform 36 6 Introduction 36 6 Transformations and the Riordan Group The Catalan transform Transforms of a Jacobsthal family The Generalized Ballot Transform The Signed Generalized Ballot transform An Associated Transformation Combining transformations 57 7 An application of Riordan arrays to coding theory 59 7 Introduction 59 7 Error-correcting codes Introducing the one-parameter family of MDS transforms Applications to MDS codes 65 8 Lah and Laguerre transforms of integer sequences 67 8 The Lah transform 68 8 The generalized Lah transform Laguerre related transforms The Associated Laguerre transforms The Generalized Laguerre transform 76 x 86 Transforming the expansion of 76 µx νx 87 The Lah and Laguerre transforms and Stirling numbers The generalized Lah, Laguerre and Stirling matrices 8 iii

4 89 Stirling numbers and Charlier polynomials 8 80 Appendix A - the Laguerre and associated Laguerre functions 83 8 Appendix B - Lah and Laguerre transforms in the OEIS 84 9 Riordan arrays and Krawtchou polynomials 86 9 Introduction 86 9 Krawtchou polynomials Krawtchou polynomials and Riordan arrays 89 0 On Integer-Sequence-Based Constructions of Generalized Pascal Triangles98 0 Introduction 98 0 Preliminaries The Narayana Triangle 99 x 04 On the series reversion of and x( ax 03 +αx+βx bx 05 Introducing the family of centrally symmetric invertible triangles 4 06 A one-parameter sub-family of triangles 0 07 The Jacobsthal and the Fibonacci cases 4 08 The general case 7 09 Exponential-factorial triangles 9 00A generalized Riordan array 36 0A note on generalized Stirling matrices 39 0Generalized Charlier polynomials 43 Generalized Pascal Triangles Defined by Exponential Riordan Arrays 45 Introduction 45 Preliminaries 45 3 Introducing the family of centrally symmetric invertible triangles 49 4 The General Case 60 5 The case r 64 6 A family of generalized Narayana triangles 67 The Hanel transform of integer sequences 69 The Hanel transform 69 Examples of the Hanel transform of an integer sequence 70 3 A family of Hanel transforms defined by the Catalan numbers 74 4 Krattenthaler s results 76 3 Row sum and central coefficient sequences of Pascal triangles defined by exponential Riordan arrays 79 3 The family B r of Pascal-lie matrices 8 3 Central sequences related to the family T r Central coefficient sequences of the family B r A note on the construction of T r 88 iv

5 4 Generalized trinomial numbers, orthogonal polynomials and Hanel transforms 89 4 Introduction 89 4 The central trinomial coefficients, orthogonal polynomials and Hanel transform90 43 Generalized central trinomial coefficients, orthogonal polynomials and Hanel transforms A conjecture On the row sums of L(α, β (a n, Pascal-lie triangles Hanel transform of generalized Catalan numbers Hanel transform of the sum of consecutive generalized Catalan numbers 304 Bibliography 309 v

6 Declaration of Authorship I, PAUL BARRY, declare that this thesis titled, A Study of Integer Sequences, Riordan Arrays, Pascal-lie Arrays and Hanel Transforms and the wor presented in it are my own I confirm that: This wor was done wholly or mainly while in candidature for a research degree at this University Where I have consulted the published wor of others, this is always clearly attributed 3 Where I have quoted from the wor of others, the source is always given With the exception of such quotations, this thesis is entirely my own wor 4 I have acnowledged all main sources of help 5 Where the thesis is based on wor done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself Signed: Date: vi

7 Abstract We study integer sequences and transforms that operate on them Many of these transforms are defined by triangular arrays of integers, with a particular focus on Riordan arrays and Pascal-lie arrays In order to explore the structure of these transforms, use is made of methods coming from the theory of continued fractions, hypergeometric functions, orthogonal polynomials and most importantly from the Riordan groups of matrices We apply the Riordan array concept to the study of sequences related to graphs and codes In particular, we study sequences derived from the cyclic groups that provide an infinite family of colourings of Pascal s triangle We also relate a particular family of Riordan arrays to the weight distribution of MDS error-correcting codes The Krawtchou polynomials are shown to give rise to many different families of Riordan arrays We define and investigate Catalannumber-based transformations of integer sequences, as well as transformations based on Laguerre and related polynomials We develop two new constructions of families of Pascallie number triangles, based respectively on the ordinary Riordan group and the exponential Riordan group, and we study the properties of sequences arising from these constructions, most notably the central coefficients and the generalized Catalan numbers associated to the triangles New exponential-factorial constructions are developed to further extend this theory The study of orthogonal polynomials such as those of Chebyshev, Hermite, Laguerre and Charlier are placed in the context of Riordan arrays, and new results are found We also extend results on the Stirling numbers of the first and second ind, using exponential Riordan arrays We study the integer Hanel transform of many families of integer sequences, exploring lins to related orthogonal polynomials and their coefficient arrays Two particular cases of power series inversion are studied extensively, leading to results concerning the Narayana triangles

8 Acnowledgements I am indebted to Professor Patric Fitzpatric of University College Cor for his encouragement in this endeavour Special thans must go to my wife, Mary, for her constant support and understanding during the writing of this wor Special thans are also due to Nadine and Peter, both of whom provided many reasons for continuing when the path was less than clear I have gained much by woring in collaboration with Prof Dr Predrag Rajović and Dr Maro Petović, of the University of Niš, Serbia It is a pleasure to acnowledge this All who wor in the area of integer sequences are completely in the debt of Neil Sloane, whose Online Encyclopedia of Integer Sequences must stand as one of the greatest achievements of a single person in modern times Gratitude is also expressed to Jeffrey Shallit, editor-in-chief of the Journal of Integer Sequences, for his continued promotion of the growing literature surrounding integer sequences

9 Chapter Introduction Overview of this wor The central object of this wor is the study of integer sequences, using both classical methods and methods that have emerged more recently, and in particular the methods that have been inspired by the concept of Riordan array A leading theme is the use of transformations of integer sequences, many of them defined by Riordan arrays In this context, a transformation that has attracted much attention in recent years stands out This is the Hanel transform of integer sequences This is not defined by Riordan arrays, but in this wor we study some of the lins that exist between this transformation and Riordan arrays This lin is determined by the nature of the sequences subjected to the Hanel transforms, and in the main, we confine ourselves to sequences which themselves are closely lined to Riordan arrays This aids in the study of the algebraic and combinatorial nature of this transform, when applied to such sequences Many of the sequences that we will study in the context of the Hanel transform are moments sequences, defined by measures on the real line This builds a bridge to the world of real analysis, and indeed to functional analysis Associated to these sequences is the classical theory of orthogonal polynomials, continued fractions, and lattice paths An important aspect of this wor is the construction of so-called Pascal-lie number arrays In many cases, we construct such arrays using ordinary, exponential or generalized Riordan arrays, which are found to give a uniform approach to certain of these constructions We also loo at other methods of construction of Pascal-lie arrays where appropriate, to provide a contrast with the Riordan array inspired constructions The plan of this wor is as follows In this Introduction, we give an overview of the wor and outline its structure In Chapter we review many of the elements of the theory of integer sequences that will be important in ensuing chapters, including different ways of defining and describing an integer sequence Preparatory ground is laid to study lins between certain integer sequences, orthogonal polynomials and continued fractions, and the Hanel transform This also includes a loo at hypergeometric series We finish this chapter by looing at different ways of defining triangular arrays of integers, some of which are simple Pascal-lie arrays Illustrative examples are to be found throughout this chapter

10 In Chapter 3, based on the published wor [9], we explore lins between the cyclic groups, integer sequences, and decompositions of Pascal s triangle The circulant nature of the associated adjacency matrices is exploited, allowing us to use Fourier analysis techniques to achieve our results We finish by looing at the complete graphs as well In Chapter 4, we review the notion of Riordan group, and some of its generalizations Examples are given that will be used in later chapters The chapter ends by looing at the notion of production matrices In Chapter 5, we briefly introduce the topic of the so-called Deleham DELTA construction This method of constructing number triangles is helpful in the sequel To our nowledge, this is the first time that this construction has been analyzed in the manner presented here In Chapter 6, based on the published article [5], we study certain transformations on integer sequences defined by Riordan arrays whose definitions are closely related to the generating function of the Catalan numbers These transformations in many cases turn out to be well-nown and important Subsequent chapters explore lins between these matrices and the structure of the Hanel transform of certain sequences In Chapter 7 we give an example of the application of the theory of Riordan arrays to the area of MDS codes This chapter has appeared as [0] In Chapter 8, based on the published paper [8], we apply the theory of exponential Riordan arrays to explore certain binomial and factorial-based transformation matrices These techniques allow us to easily introduce generalizations of these transformations and to explore some of the properties of these new transformations Lins to classical orthogonal polynomials (eg, the Laguerre polynomials and classical number arrays are made explicit In Chapter 9 we continue to investigate lins between certain Riordan arrays and orthogonal polynomials We also study lins between exponential Riordan arrays and the umbral calculus This chapter has appeared as [] In Chapter 0 we use the formalism of Riordan arrays to define and analyze certain Pascal-lie triangles Lins are drawn between sequences that emerge from this study and the reversion of certain simpler sequences We finish this chapter by looing at alternative ways of constructing Pascal-lie triangles, based on factorial and exponential methods In this section we introduce and study the notion of sequence-specific generalized exponential arrays An earlier version of this chapter has appeared as [6] In Chapter we continue the exploration of the construction of Pascal-lie triangles, this time using exponential Riordan arrays as the medium of construction In the final section we briefly indicate how some of the methods introduced in the final section of Chapter 0 can be used to build a family of generalized Narayana triangles An earlier version of this chapter has appeared as [7] In Chapter we give a brief introduction to the theory of the Hanel transform of integer sequences, using relevant examples to prepare the ground for further chapters In Chapter 3 we extend the study already commenced in Chapter, and we also loo at the Hanel transforms of some of the sequences that emerge from this extension In Chapter 4 we calculate the Hanel transform of sequences related to the central trinomial coefficients, and we conjecture the form of the Hanel transform of other associated sequences Techniques related to Riordan arrays and orthogonal polynomials are used in 3

11 this chapter Elements of this chapter have been presented at the Applied Linear Algebra (ALA008 conference in honour of Ivo Mare, held in the University of Novi Sad, May 008 A forthcoming paper based on this in collaboration with Dr Predrag Rajović and Dr Maro Petović has been submitted to the Journal of Applied Linear Algebra The author wishes to acnowledge what he has learnt through collaborating with Dr Predrag Rajović and Dr Maro Petović, both of the University of Niš, Serbia This collaboration centred initially on Hanel transform methods first deployed in [6], and subsequently used in [88], as well as in the chapters concerning the calculation of the Hanel transform of integer sequences 4

12 Chapter Preliminary Material Integer sequences We denote by N the set of natural numbers N {,, 3, 4, } When we include the element 0, we obtain the set of non-negative integers N 0, or N 0 {0,,, 3, 4, } N 0 is an ordered semigroup for the binary operation + : N 0 N 0 N 0 N 0 is a subset of the ring of integers Z obtained from N 0 by adjoining to N 0 the element n for each n N, where n is the unique element such that n + ( n 0 By an integer sequence we shall mean an element of the set Z N 0 Regarded as an infinite group, the set Z N 0 is called the Baer-Specer group [57, 99] Thus a (one-sided integer sequence a(n is a mapping a : N 0 Z where a(n denotes the image of n N 0 under this mapping The set of such integer sequences Z N 0 inherits a ring structure from the image space Z Thus two sequences a(n and b(n define a new sequence (a + b(n by the rule (a + b(n a(n + b(n, and similarly we obtain a sequence (ab(n by the rule (ab(n a(nb(n The additive inverse of the sequence a(n is the sequence with general term a(n An additional binary operation, called convolution, may be defined on sequences as follows: (a b(n a(b(n 5

13 We then have a b(n b a(n In addition, the sequence δ n 0 n (, 0, 0, 0, plays a special role for this operation, since we have a δ(n a(n for all n A related binary operation is that of the exponential convolution of two sequences, defined as n ( n a(b(n Frequently we shall use the notation a n for the term a(n For a sequence a n, we define its binomial transform to be the sequence b n ( n a This transformation has many interesting properties, some of which will be examined later In the sequence, we shall use the notation B to denote the matrix with general term ( n Integer sequences may be characterized in many ways In the sequel, we shall frequently use the following methods: Generating functions Recurrences 3 Moments 4 Combinatorial definition We shall examine each of these shortly The On-Line Encyclopedia of Integer Sequences Many integer sequences and their properties are to be found electronically on the On-Line Encyclopedia of Sequences [05, 06] Sequences therein are referred to by their A number, which taes the form of Annnnnn We shall follow this practice, and refer to sequences by their A number, should one exist 3 Polynomials We let R denote an arbitrary ring Let x denote an indeterminate Then an expression of the form P (x a x, where a i R for 0 i n is called a polynomial in the unnown x over the ring R If a n 0 then n is called the degree of the polynomial P 6

14 We denote by R[x] the set of polynomials over the ring R The set of polynomials over R inherits a ring structure from the base ring R For instance, if P, Q R[x], where P (x and Q(x then we define P + Q R[x] as the element n P n Q i0 a x b i x i, (P + Q(x max(n P,n Q j0 (a j + b j x j, where we extend either the a or the b i by zero values as required A polynomial P (x n P a x is called monic if the coefficient of the highest order term is A polynomial sequence with values in R[x] is an element of R[x] N 0 An example of an important sequence of polynomials is the family of Chebyshev polynomials of the second ind n ( n U n (x ( (x n The Chebyshev polynomials of the first ind (T n (x n 0 are defined by T n (x n + 0n n The Bessel polynomials y n (x are defined by ( n + 0 n ( n (x n y n (x (n +!!(n! x (see [08] The reverse Bessel polynomials are then given by Θ n (x (n +!!(n! xn 4 Orthogonal polynomials By an orthogonal polynomial sequence (p n (x n 0 we shall understand [53, 99] an infinite sequence of polynomials p n (x, n 0, with real coefficients (often integer coefficients that 7

15 are mutually orthogonal on an interval [x 0, x ] (where x 0 is allowed, as well as x, with respect to a weight function w : [x 0, x ] R : x p n (xp m (xw(xdx δ nm hn h m, x 0 where x x 0 p n(xw(xdx h n We assume that w is strictly positive on the interval (x 0, x Every such sequence obeys a so-called three-term recurrence : p n+ (x (a n x + b n p n (x c n p n (x for coefficients a n, b n and c n that depend on n but not x We note that if then p j (x j x j + jx j + j 0,, a n ( ( n+, b n a n+ n n n h n, c n a n n n+ n 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 diagonal elements of this array will all be In this case, we can write the three-term recurrence as p n+ (x (x α n p n (x β n p n (x, p 0 (x, p (x x α 0 The moments associated to the orthogonal polynomial sequence are the numbers µ n x x 0 x n w(xdx We can find p n (x, α n and β n from a nowledge of these moments To do this, we let n be the Hanel determinant µ i+j n i,j 0 and n,x be the same determinant, but with the last row equal to, x, x, Then p n (x n,x n ( u u More generally, we let H be the determinant of Hanel type with (i, j-th v v term µ ui +v j Let n H ( 0 n 0 n ( 0 n n, H 0 n n + Then we have α n n n n n, 8 β n n n n

16 Given a family of monic orthogonal polynomials 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 p n (x a n, x Then we have n+ n a n+, x (x α n a n, x β n a n, x from which we deduce a n+,0 α n a n,0 β n a n,0 ( and a n+, a n, αa n, β n a n, ( 5 Power Series Again, we let R denote an arbitrary ring An expression of the form p(x a x, is called a (formal power series in the indeterminate x a is called the -th coefficient of the power series We denote by R[[x]] the set of formal power series in x over the ring R [0] R(x is a ring For instance, if q(x b i x i, then we can define the sum of p and q as (p + q(x (a j + b j x j j0 Example We consider the power series x Here, the -th coefficient of the power series is If for instance x C is a complex number with x <, then it is nown that x x 9

17 6 Ordinary generating functions For a sequence a n, we define its ordinary generating function (ogf to be the power series f(x a n x n Thus a n is the coefficient of x n in the power series f(x We often denote this by a n [x n ]f(x Example The sequence 0 n The sequence with elements, 0, 0, 0, has ogf given by f(x Example 3 The sequence The sequence with elements,,,, has ogf f(x which we can formally express as x We shall on occasion refer to this as the sequence ( n or just n We note that we have ( n 0 Thus the binomial transform of 0 n is n Similarly, ( n ( n 0 ( n Thus the inverse binomial transform of 0 n is ( n In general, we have the following chain of binomial transforms : corresponding to the generating functions ( n ( n 0 n n n + x + x 0x x x Example 4 Fibonacci numbers with defining recurrence x The Fibonacci numbers 0,,,, 3, 5, 8, A F n F n + F n, F 0, F, have ogf x x x 0

18 Example 5 Jacobsthal numbers The Jacobsthal numbers J n J(n n 3 ( n 3 A00045 have generating function x x x They begin 0,,, 3, 5,,, The sequence J (n n 3 x x x + ( n has ogf This sequence begins, 0,,, 6, 0,, A The sequence with elements J(n + + J (n form the Jacobsthal-Lucas sequence A0455 This sequence has ogf given by x x x If A(x, B(x and C(x are the ordinary generating functions of the sequences (a n,(b n and (c n respectively, then A(x B(x if and only if a n b n for all n Let λ, µ Z, such that c n λa n + µb n for all n Then C(x λa(x + µb(x 3 If c a b then C(x A(xB(x and vice versa If the power series f(x a x is such that a 0 0 (and hence f(0 0, then we can define the compositional inverse f(x of f to be the unique power series such that f( f(x x f is also called the reversion of f We shall use the notation f Revf for this We note that necessarily f(0 0 Example 6 The generating function has compositional inverse f given by f(x f(x x x x + x + 5x x x This is obtained by solving the equation u u u x where u u(x f(x We note that the equation u u u x has two formal solutions; the one above, and + x + 5x + x + ũ x We reject this solution as it does not have a power series expansion such that ũ(0 0

19 7 Exponential generating functions For a sequence (a n n 0, we define its exponential generating function (egf to be the power series a n f(x n! xn n0 In other words, f(x is the ogf of the sequence ( a n n! Example 7 exp(x e x is the egf of the sequence,,, Example 8 cosh(x is the egf of the sequence, 0,, 0,, 0, with general term Example 9 x is the egf of n! ( + ( n If A(x, B(x and C(x are the exponential generating functions of the sequences (a n,(b n and (c n respectively, then A(x B(x if and only if a n b n for all n Let λ, µ Z, such that c n λa n + µb n for all n Then C(x λa(x + µb(x 3 If c n n ( n a b n then C(x A(xB(x and vice versa Example 0 The Bessel function I 0 (x is the egf of the aerated central binomial numbers, 0,, 0, 6, 0, 0, 0, 70, with general term ( n a n n ( + ( n / Then the product exp(xi 0 (x is the egf of the sequence t n ( n a ( n a since exp(x is the egf of the sequence b n t n is the sequence,, 3, 7, 9, 5, 4, of central trinomial numbers, where t n coefficient of x n in ( + x + x n We note that for n m +, the expression ( n n has the value Γ(m + Γ ( m+3

20 8 Generalized generating functions We follow [6] in this section Given a sequence (c n n 0, the formal power series f(t f t c is called the generating function with respect to the sequence c n of the sequence (f n n 0, where c n is a fixed sequence of non-zero constants with c 0 In particular, f(t is the ordinary generating function if c n for all n, and f(t is the exponential generating function if c n n! 9 The Method of Coefficients The method of coefficients [58, 57] consists of the consistent application of a set of rules for the functional [x n ] : C[[x]] C In the sequel, we shall usually wor with the restriction [x n ] : Z[[x]] Z For f(x and g(x formal power series, the following statements hold : [x n ](αf(x + βg(x α[x n ]f(x + β[x n ]g(x K (linearity [x n ]xf(x [x n ]f(x K (shifting [x n ]f (x (n + [x n+ ]f(x K3 (differentiation [x n ]f(xg(x ([y ]f(y[x n ]g(x K4 (convolution [x n ]f(g(x ([y ]f(y[x n ]g(x [x n ] f n [xn ] We note that the rule K3 may be written as K5 (composition ( n x K6 (inversion f(x [t n ]f(t n [tn ]f (t Example We extend the following result : ( n n F n+ + F n n of [58] This identity is based on two facts : ( log t t 3

21 and the following identity, a consequence of the rules for the evaluation of Riordan arrays : ( m + a f [t n ]( + t m f(t b ( + t a (b < 0 n + b We use the following fact We now wish to evaluate the expression [t n ] αt βt n ( n α n β (3 n+ ( α β n n n We have n+ ( α β n n βn n+ [ β n [t n ] β n [t n ] ln βn βn n ln ( n ( α β ( αy β ( α β t α β t α( + t n [tn β ] αt α β β t α + t β [tn ] αt α β β t n y t( + t ] Thus using Eq (3, we obtain n+ ( n n α β n n ( n n ( n α n β + α n β + For α β, we retrieve the Fibonacci result above For α, β, we obtain that that for n > 0, we have n+ the Jacobsthal-Lucas numbers (A0455 ( n n n J(n + + J (n 4

22 0 Lagrange inversion Let F C[[t]] If f(t F with f(t f t and r is the minimum integer for which f r 0, then r is called the order of f(t The set of formal power series of order r is denoted by F r F 0 is the set of invertible formal power series, that is, series f(t for which a series f (t exists in F such that f(tf (t One version of Lagrange inversion [57] is given by rule K6: [t n ] f ( n t n [tn ] f(t If now we have where φ F 0, then if we define f by we have f w and so w(t tφ(w(t f(y y φ(y, [t n ]w(t ( n t n [tn ] f(t n [tn ]φ(t n Now let F F, and let w(t tφ(w(t Then [t n ]F (w(t n [tn ]F (tφ(t n Also, we have, for F, φ F, [ ] [t n ]F (tφ(t n [t n F (w ] w tφ(w tφ (w Example Generalized central trinomial coefficients We wish to find the generating function of [t n ]( + αt + βt n, the central trinomial coefficients (for the parameters α, β We let F (t, and φ(t + αt + βt We have w t( + αw + βw, and so w αt αt + (α 4βt βt Thus [ ] [t n ]( + αt + βt n [t n F (w ] w tφ(w tφ (w [ ] [t n αt αt + (α 4βt ] w t(α + βw βt [ [t n ] αx + (α 4βx w αt ] αt + (α 4βt βt 5

23 This shows that the required generating function of the generalized central trinomial coefficients is given by αx + (α 4βx Example 3 The Riordan array (, xc(x Anticipating the developments of Chapter 4, we see to calculate the general terms of the Riordan array (, xc(x Now Thus we let and so w tφ(w implies (, xc(x (, x( x φ(w w w( w t We also let F (t t and so F (t t Then [t n ](w(t [t n ]F (w(t n [tn ]F (tφ(t n n ( n [tn ]t t ( n n [tn ]t i i0 ( n + i n [tn ] i i0 ( n + n n n ( n n n ( t i t i+ Adjusting for the first row, we obtain that the general term of the Catalan array (, xc(x is given by + 0 n n + 0 n ( n n Recurrence relations Recurrence relations allow us to express the general term of a sequence as a function of earlier terms Thus we may be able to express the term a n as a function of a 0, a,, a n for all n r r is called the order of the recurrence The values a 0, a,, a r are called the initial values of the recurrence 6

24 Example 4 The sequence defined by the recurrence a n a n + a n with initial values a 0 0, a is the Fibonacci sequence A given by (( a n F (n n ( n It is easy to calculate the ogf of this sequence Letting A(x n0 a nx n, and multiplying both sides of the recurrence by x n and summing for n, we find that a n x n n a n x n + n a n x n n Now while, for instance, a n x n a n x n a x a 0 A(x x, n n0 Thus we obtain a n x n x a n x n x(a(x a 0 xa(x n n A(x x xa(x + x A(x or x A(x x x Thus the generating function of the Fibonacci numbers is Example 5 The sequence defined by the recurrence a n a n + a n x x x with initial values a 0 0, a is the Jacobsthal sequence A00045 given by a n J(n n 3 ( n 3 This sequence starts 0,,, 3, 5,,, The generating function of the Jacobsthal numbers is x x x In the above two examples, the recurrence was linear, of order The following example, defining the well-nown Catalan numbers A00008, is of a different nature 7

25 Example 6 The sequence defined by the recurrence n C n C i C n i i0 with C 0 is the sequence of Catalan numbers, which begins,,, 5, 4, 4, sequence has been extensively studied and has many interesting properties The generating function of the Catalan numbers is the function This c(x 4x x We note that the series reversion of xc(x is given by x( x One way to see this is to solve the equation uc(u x We do this with the following steps 4u x 4u x 4u x Moment sequences 4u ( x 4x + 4x 4u 4x 4x u x( x Many well-nown integer sequences can be represented as the moments of measures on the real line For example, we have ( n n C n π C n+ π ( n n π + ( n π x(4 x x n dx, x x n x(4 xdx, x n x(4 x dx, x n 4 x dx It is interesting to study the binomial transform of such a sequence If the sequence a n has the moment representation a n β α x n w(xdx 8

26 then we have b n ( n a ( n β x w(xdx α x w(xdx β α β α ( + x n w(xdx Note that the change of variable y x + gives us the alternative form b n β α ( + x n w(xdx β+ α+ y n w(y dy Example 7 The central trinomial numbers t n [x n ](+x+x n are given by the binomial transform of the aerated sequence ( n +( n n Thus t n π π 3 ( + x n 4 x dx x n 3 + x x dx The r-th binomial transform of a n is similarly given by More generally, we have β α β α (r + x n w(xdx (r + sx n w(xdx ( n r n s a Aspects of these general binomial transforms have been studied in a more general context in [07] Example 8 We consider the sequence, 3,, 5,, 978, or A with ogf and general term 3 4x (( ( n n n n ( + n n + + n 9

27 We have a n 3 4 π 3 π x n 4 x 9 x x dx + ( 9 x n 4 x 9 x x dx + δ 9 n, x n We note that this sequence is the image of n by the Riordan array (c(x, xc(x Thus a n is defined by a so-called Sobolev measure [50] Example 9 The sequence, 0,, 0, 3, 0, 5, 0, 05, 0, 945, with general term a n ((n/!! + ( n where the double factorials (n!! n ( is A0047, counts the number of perfect matchings in K n, the complete graph on n vertices We have [04] The egf of this sequence is ex Note that we have (n!! π a n π The binomial transform b n of a n is given by b n 0 x n e x dx x n e x x dx π ( + x n e x dx π π x n e (x dx, x n e x dx which is A This counts, for instance, the number of Young tableaux with n cells We note that the Hanel transform of this last sequence is given by (n n n!! This is A08400 Anticipating Chapter 4 we can represent A0047 as the row sums of the exponential Riordan array [ e x, x( + x] which begins

28 This is A The Stieltjes transform of a measure The Stieltjes transform of a measure µ on R is a function G µ defined on C \ R by G µ (z z t µ(t If f is a bounded continuous function on R, we have f(xµ(x lim f(xig µ (x + iydx y 0 + R If µ has compact support, then G µ is holomorphic at infinity and for large z, G µ (z R R n0 a n z n+, where a n R tn µ(t are the moments of the measure If µ(t dψ(t ψ (tdt then ψ(t ψ(t 0 π lim y 0 + t t 0 IG µ (x + iydx If now g(x is the generating function of a sequence a n, with g(x n0 a nx n, then we can define G(z ( z g a n z z n+ By this means, under the right circumstances we can retrieve the density function for the measure that defines the elements a n as moments 4 Orthogonal polynomials as moments Many common orthogonal polynomials, suitably parameterized, can be shown to be moments of other families of orthogonal polynomials This is the content of [9, 0] This allows us to derive results about the moment sequences in a well-nown manner, once the characteristics (for instance, the three term recurrence relation of the generating family of orthogonal polynomials are nown Such characteristics of common orthogonal polynomials may be found in [6] This approach has been emphasized in [3], for instance, in the context of the evaluation of the Hanel transform of sequences Example 0 A simple example [, 34] of this technique is as follows The reversion of the generating function generates the sequence with general term x +αx+βx u n n ( n n0 C α n β

29 (see Chapter 0 We are interested for this example in the Hanel transform of u n+ For this, we cast u n+ into hypergeometric form : ( u n+ α n F n, n ; 4β α Applying the transformation F ( α, + α; + β; z we obtain ( ( z α F α, β; β; u n+ (α ( β n F n, 3 ; 3; 4 β β α z, z This exhibits u n+ as a Meixner polynomial Meixner polynomials are moments for the Jacobi polynomials [9, 5] Hence we can readily compute the Hanel determinant of u n+ (it is equal to (α(α β (n+ 5 Lattice paths Many well-nown integer sequences can be represented by the number of paths through a lattice, where various restrictions are placed on the paths - for example, the types of allowable steps The best-nown example is the Catalan numbers, which count Dyc paths in the plane Lattice paths can be defined in two distinct but equivalent ways - explicitly, as a sequence of points in the plane, or implicitly, as a sequence of steps of defined types (we can find the points in the plane by following the steps Thus we can thin of a lattice path [43] as a sequence of points in the integer lattice Z, where a pair of consecutive points is called a step of the path A valuation is a function on the set of possible steps Z Z A valuation of a path is the product of the valuations of its steps Alternatively, given a subset S of Z Z we can define a lattice path with step set S to be a finite sequence Γ s s s where s i S for all i [56] Well nown and important paths include Dyc paths, Motzin paths and Schröder paths Example A Dyc path is a path starting at (0, 0 and ending at (n, 0 with allowable steps (, (a rise and (, (a fall, which does not go below the x-axis Thus S {(,, (, } Such paths are enumerated by the Catalan numbers C n The central binomial coefficients ( n n count all such paths, when the restriction of not going below the x-axis is lifted (such paths are then called Grand-Dyc paths or binomial paths [78] Example The n-th central binomial coefficient, [x n ]( + x + x n, counts the number of lattice paths starting at (0, 0 and ending at (n, 0, whose allowed steps are (, 0, (, and (, Thus in this case S {(, 0, (,, (, } The Motzin numbers m n n ( n ( F ( n, n ; ; 4 count the number of such paths that do not descend below the x-axis A Dyc path is clearly a special case of a Motzin path

30 Example 3 A Schröder path is a path that starts at (0, 0, ends at (n, 0, and has allowable steps (,, (, 0 and (, The Schröder numbers S n F ( n, n+; ; count the number of such paths that do not go below the x-axis Paths may be coloured, that is, for each step s S, we can assign it an element from a finite set of colours Families of disjoint paths play an important role in the evaluation of certain important determinants, including Hanel determinants [3] For instance in the case [5] of the Catalan numbers C n, if we define H n ( C +i+j 0 i,j n then this determinant is given by the number of n-tuples (γ 0,, γ n of vertex-disjoint paths in the integer lattice Z Z (with directed vertices from (i, j to either (i, j+ or to (i+, j never crossing the diagonal x y, where the path γ r is from ( r, r to ( + r, + r 6 Continued fractions Continued fractions [7] play an important role in many areas of combinatorics They are naturally associated to orthogonal polynomials and lattice path enumeration [89] They play an important role in the computation of Hanel transforms In this section we briefly define continued fractions and give examples of their application to integer sequence A generalized continued fraction is an expression of the form t b 0 + b + b + a a a 3 b 3 + a 4 where the a n (n > 0 are the partial numerators, the b n are the partial denominators, and the leading term b 0 is the so-called whole or integer part of the continued fraction The successive convergents (also called approximants of the continued fraction are formed as follows : t 0 A 0 B 0 b 0, t A B b b 0 + a b, t A B b (b b 0 + a + a b 0 b b + a, where A n is the numerator and B n is the denominator (also called continuant of the nth convergent, and where we have the following recurrence relations : A, B 0, A 0 b 0, B 0 ; for p 0,,, A p+ b p+ A p + a p+ A p, B p+ b p+ B p + a p+ B p 3

31 A n B n is called the nth convergent (approximant We have A n B n b 0 + b + b + b 3 + a a a 3 a 4 + a n b n The convergents of a continued fraction do not change when an equivalence transformation is effected as follows: Example 4 c a b 0 + c c a c b + c c 3 a 3 c b + c 3 b 3 + c 3c 4 a 4 c 4 b 4 + c(x xc(x 4x x x x is the generating function of the Catalan numbers The denominator polynomials are then given by,, x, x, 3x + x, 4x + 3x, 5x + 6x x 3, Thus the nth denominator polynomial is given by n ( n B n (x ( x The binomial transform of the Catalan numbers has generating function x x x x x x x x 4

32 The continued fraction mx mx x x x x mx x is the generating function of the m-th binomial transform of the Catalan numbers (ie it is equal to c ( x mx mx More generally, we have rxc(x rx x x The denominator polynomials are then given by,, rx, (r + x, (r + x + rx, (r + 3x + (r + x, Thus the nth denominator polynomial is given by n (( n B n (x ( ( n + The Hanel transform of the sequence with gf is rxc(x rn We have c(rx rx rx rx r x c(rx is the gf of the sequence r n C n which has Hanel transform r n(n+ Example 5 The continued fraction g(x; r rx x rx x rx 5

33 generates the sequence a n (r which begins, r, r(r +, r(r + 3r +, r(r 3 + 6r + 6r +, r(r 4 + 0r 3 + 0r + 0r +, which is the Narayana transform (see Example 84 of the power sequence, r, r, r 3, r 4, : r r r r(r r 3 r(r + 3r +, r 4 r(r 3 + 6r + 6r r 5 r(r 4 + 0r 3 + 0r + 0r + a n (r N(n, r For r 0,,, we obtain the following sequences r A r A00008 r A00638 r A04789 r A0898 r A0830 which include the Catalan numbers (r and the large Schröder numbers (r These sequences can be characterized as a n (r [x n+ x( x ]Rev + (r x The sequence a n (r has Hanel transform r (n+ Example 6 g(x x x 3x 4x is the gf of the double factorials (n!! n (n! (, whose egf is n n! x The denominator polynomials are then given by,, x, 3x, 6x + 3x, 0x + 5x, 5x + 45x 5x 3, 6

34 Thus the nth denominator polynomial is given by Example 7 n ( n B n (x ( +!!( x x x x x 3x is the gf of the factorial numbers n! with egf given by x The denominator polynomials are,, x, x, 4x + x, 6x + 6x, 9x + 8x 6x 3, x + 36x 4x 3, We obtain the following array of numbers as the coefficient array of these polynomials : The second column is minus times the quarter squares n n A0060 The third column is given by n ( n We note that this last sequence appears to be twice A0004(n +, where A0004 gives the crossing number of K n, the complete graph with n nodes The formula appears to be consistent with Zaraniewicz s conjecture (which states the the graph crossing number of the complete bigraph K n,m is given by n n m m [47] In fact (see below, we conjecture that A0004 is given by ( n ( n An alternately signed version of every second row of this array is given by

35 This is A000, the triangle of coefficients of the Laguerre polynomials L n (x (powers of x in decreasing order It has general term T ( n, ( n! ( n Taing the second embedded triangle of alternate rows, we obtain which has general term T ( n, ( (n +! ( n (n +! We note that the quotient is the Narayana triangle (see A Combining the terms for T ( n, polynomials is given by and T ( n, T ( n, ( ( +! ( ( n + ( n! we see that the coefficient array for the denominator ( T n, ( n+ ( n! This triangle is A458 We note that the triangle with general term ( ( n+ n is A448 This triangle has row sums equal to ( n A variant is given by A04559, which n counts the number of left factors of pealess Motzin paths of length n having number of U s and D s Example 8 The Bell numbers A0000 have gf given by x 8 x x x x 3x x

36 The bi-variate generating function generates the array that starts xy x x x x x x with row sums equal to the Bell numbers A0000 This is the array (see Chapter 5 for notation [0,,,,, 3,, 4,, ] [, 0, 0, 0, ] Of particular interest for this wor is the notion of J-fraction We shall consider these in the context of a sequence c 0, c, such that H n c i+j 0 i,j n 0, n 0 Then there exists a family of orthogonal polynomials P n (x that satisfy the recurrence P n+ (x (x α n P n (x β n P n (x This means that the family P n (x are the denominator polynomials for the J-fraction β x α 0 x β x α x α x The theory now tells us that in fact g(x c n x n c 0 β n0 x α 0 x β x α x α x 9

37 is the generating function of the sequence In addition, the Hanel transform of c n is then given by n h n [7, 3, 3] Example 9 The continued fraction β n + x x x x x x x is the J-fraction generating function of the Catalan numbers More generally, we have c(x xc(x ( + x x x x x x x This is therefore the image of the power sequence n (with gf under the Riordan x array (c(x, xc(x Each of these sequences has Hanel transform h n Thus ( + n r [x n ] n + n x (r + x x x x x x We note that the gf of C n+ is given by x while that of ( n+ n A00700 is given by 3x x x 30 x x x x x x x x

38 The sequence with gf given by is Fine s sequence A x x x x x Example 30 The sequence A0667 which begins, 3,, 43, 73,, with gf 4x x 3x x x x x x has Hanel transform n The sequence,, 3,, 43, 73, which has gf x 3x x x x x x x also has Hanel transform n A0667 is a transform of F (n + by the Riordan array (, xc(x The sequence,, 3,, 43, 73, is the image of A0059, or F (n, by (, xc(x Example 3 The sequence, 5, 8, 6, 934, 5438, with gf 5x x 3x x x x x x is the image of A07839, or [x n ] under the Riordan array (, xc(x It has Hanel 5x+x transform 3 n Looing now at the sequence,, 5, 8, 6, 934, 5438, we find that it has 3

39 a gf given by x 4x 9 4 x 5 6 x 4 4x 5 x x 36 x x 49 x 4 This sequence has Hanel transform (n + 33 n with gf x ( 3x (A00634 Example 3 In this example we characterize a family of sequences in a ( number of ways The family, parameterized by r, is obtained by applying the Riordan array to x, x( x rx rx the Catalan numbers C Thus let a n (r denote the r-th element of this family We have a n (r with generating function + j0 ( j ( + j In terms of continued fractions we find that ( n j n j g(x; r x ( x( x rx c rx g(x; r rx, x x rx x r n j C, where the coefficients follow the pattern r,,, r,,, r,,, As a J-fraction, we have g(x; r rx rx rx x (r + x x (r + x rx x Here, the α sequence is r,, r+, r+,, r+, r+,, while the β sequence is r, r,, r, r,, r, r,, (starting at β The Hanel transform of a n (r is then given by h n r (n+ 3 a n ( is A05979 with Hanel transform A3475 3

40 Example 33 The continued fraction x x x x 3x 3x 4x is the generating function of the Bell numbers (see also Example 8, which enumerate the total number of partitions of [n] These are the numbers,,, 5, 5, 5, 03, A0000 with egf e ex They satisfy the recurrence a n+ n ( n a From the above, we have that n h n n + which is,,,, 88, 34560, or A00078, the superfactorials Example 34 The continued fraction x x x x x 3x 4x is the generating function of the Bessel numbers, which count the non-overlapping partitions of [n] These are the numbers,,, 5, 4, 43, 43, 509, A They have Hanel transform h n Example 35 The continued fraction x x x x 3x x x is the generating function of the sequence I n of involutions, where an involution is a permutation that is its own inverse These numbers start,,, 4, 0, 6, 76, A with egf e x(+x/ Once again the Hanel transform of this sequence is given by the superfactorials Example 36 The continued fraction x 0x x 0x x 0x 0x 33

41 or x x x 0x is equal to c(x, the generating function of the aerated Catalan numbers, 0,, 0,, 0, 5, 0, 4, 0, Since β n, the Hanel transform is h n The coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( + x, x + x We note that the aerated Catalan numbers are given by ( x [x n+ ]Rev + x Example 37 The continued fraction x x x x x x x is the generating function M(x of the Motzin numbers n ( n C This is the binomial transform of the last sequence (we note that the coefficients α n are incremented by The Hanel transform of the Motzin numbers is given by h n The coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( Example 38 The continued fraction + x + x, x + x + x x 0x x 0x x 0x 0x is the generating function of the aeration of the central binomial numbers ( n n Thus these are the numbers ( n n/ (+( n / beginning, 0,, 0, 6, 0, with gf 4x Their Hanel transform is governed by the β-sequence,,,,, and hence we have h n n 34

42 The coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( x + x, x + x Example 39 The central trinomial numbers,, 3, 7, 9, A0046, or t n [x n ]( + x + x n, are given by the binomial transform of the last sequence Their generating function is then x 3x x x x x x x x The Hanel transform is again h n n The coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( x + x + x, x + x + x Example 40 The sequence,,, 3, 6, 0, 0, A00405 of central binomial coefficients ( n n has the following continued fraction expression for its generating function x x x 0x x 0x 0x Thus h n The coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( x + x, x + x Example 4 We have the following general result : If the α sequence is given by α, 0, 0, 0, 0, (ie α n α0 n, and the β sequence is given by 0, β, γ, γ, γ, (ie β 0 0, β β, β n γ for n >, then the coefficient array for the associated orthogonal polynomials (denominator polynomials is given by the Riordan array ( αx (β γx + γx, x + γx 35

43 In particular, γ γ β αγx + β 4γx βx αx γx 0x γx 0x 0x In this case we have Example 4 The continued fraction ax is the generating function of the sequence It is equal to h n β n γ (n ax bx bx bx n ( n C a n b ( bx ax c ax bx ax bx where c(x is the generating function of the Catalan numbers This sequence represents the diagonal sums of the triangular array with general term ( n C a n b To prove the initial assertion, we can proceed as follows n of applying the matrix with general term ( n a n ( n C a n b is the result to the scaled Catalan numbers b n C n Now the matrix with general term ( n a n is the generalized Riordan array ( ax, x ax 36

44 while the gf of b n C n is expressible as the continued fraction bx bx Anticipating ( results from the theory of Riordan arrays, we can thus say that the gf of n C a n b is expressible as ax b x x Simplifying this expression leads to the desired result The Hanel transform of this sequence is b (n+ If a b, we obtain the sequence n ( n C which is A This enumerates the number of Motzin paths of length n with no level steps at odd levels It represents the diagonal sums of the array ( n C, A Thus its generating function is given by ( This is equal to c x equal to x x x x b x x x x x x The first differences of this sequence have gf c x x x x x x x x x ( x x which is x x This sequence enumerates the number of Motzin paths of length n with no level steps at even levels This is A48 37

45 7 Hypergeometric functions A generalized hypergeometric function [98, 4, 04] p F q (a,, a p ; b,, b q ; x is a power series c x which can be defined in the form of a hypergeometric series, ie, a series for which the ratio of successive terms can be written c + c P ( Q( ( + a ( + a p ( + b ( + b q ( + We have pf q (a,, a p ; b,, b q ; x 0 (a (a (a p z (b (b (b q!, where (a a(a + (a + (a + We note that p F q (a,, a p ; b,, b q ; 0 For the important case p, q, we have F (a, b; c; z (a (b (c z! Example 43 F (, ; ; x x, the ogf of the sequence u n and the egf of the sequence u n n! Example 44 F (, ; ; x F (, ; ; x ( x, the ogf of the sequence u n n +, and the egf of (n +! Example 45 More generally F (a, ; ; x ( x a while F ( a, b; b; x ( + x a Example 46 We have F (/, ; ; 4x 4x, the ogf of ( n n This has Hanel transform n F (/, /; /; 4x F ( /, ; ; 4x 4x, the ogf of (n n This has n Hanel transform (n + ( n 5x x 3 F (/, /; /; 4x/( x is the ogf of the binomial transform of n ( ( n x This has Hanel transform with ogf which gives the +6x+4x sequence ( n L(n +, where L(n is the Lucas sequence A00003 x 4 F (/, ; ; 4x/( x is the binomial transform of n 5x ( n ( This has Hanel transform A0876, the trinomial transform of the Fibonacci numbers, with x ogf 6x+4x 5 F (/, ; ; 4x/(+x is the inverse binomial transform of the partial sums +x 3x of the central binomial coefficients This has Hanel transform with ogf ( x/( x + 4x or A

46 Example 47 We have F (/, ; ; 4x c(x, the ogf of the Catalan numbers In general, F ( /, ; ; 4x c(x F (/, ; ; 4x/( x x 6x+5x, the ogf of A00 The Binomial transform has ogf and the Hanel transform of A00 is F (n + x c(x +x 3 F (/, ; ; 4x/(+x +x x 3x, the ogf of A08646 The Hanel transform x x of this sequence has ogf It is the inverse binomial transform of the partial x+x sum of the Catalan numbers ie the inverse binomial transform of c(x x Example 48 We have the following: 3 F (/, ; 3; 4x is the ogf of the super-ballot numbers A007054, or 4C n C n+ [0] The Hanel transform of this sequence is n + 3 6(n! n!(n+! 3 F (/, ; 3; 4x is the ogf of C(n++C(n, or A03869 The Hanel transform of this sequence has ogf (A x 4x+x 3 0 F (/, ; 4; 4x is the sequence A0077 with general term 60(n! It has Hanel n!(n+3! transform A000447(n + with ogf 0 5x+4x x 3 ( x F (/, ; 4; 4x yields a sequence with Hanel transform with ogf ( x+x ( 3x+x /x 5 0 F (/, 3; 4; 4x is the ogf of the sequence with general term 30(n n Its Hanel n+3 0 5x+4x transform has ogf given by x 3 4x+6x 4x 3 +x 4 Example 49 F (3, ; ; 9x ( 9x /3 is the ogf 3 n n! n 3 + of A with general term Example 50 F (/, ; ; 4x( x is the ogf of the sequence [x n ](+x ( x n Example 5 We have F ( n, /; ; 4 [x n ]( + ( + x + x n n![x n ]e (+x I 0 (x [x n ] ( + x + (4 + x The coefficient array of this family of polynomials in is the array with general term ( ( n T n, 39