On the derivatives 2 P ν (z)/ ν 2 and Q ν (z)/ ν of the Legendre functions with respect to their degrees
|
|
- Kerrie Goodwin
- 5 years ago
- Views:
Transcription
1 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, 07 VOL. 8, NO. 9, Research Article On the derivatives / and / of the Legendre functions with respect to their degrees Radosław Szmytkowski Atomic and Optical Physics Division, Department of Atomic, Molecular and Optical Physics, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gdańsk, Poland ABSTRACT We provide closed-form expressions for the degree-derivatives [/ ] ν=n and [/] ν=n, with z C and n N 0, where P ν (z) and Q ν (z) are the Legendre functions of the first and the second kind, respectively. For [/ ] ν=n, we find that = P n (z) Li + B n (z) ln z + + C n (z), where Li [()/] is the dilogarithm function, P n (z) is the Legendre polynomial, while B n (z) and C n (z) are certain polynomials in z of degree n. For [/] ν=n and z C \ [, ], we derive = P n (z) Li + B n(z) ln z + π P n(z) ln z + ln z ( )n B n ( z) ln z 6 P n(z) + C n(z) ( )n C n ( z). A counterpart expression for [ Q ν (x)/] ν=n, applicable when x (, ), is also presented. Explicit representations of the polynomials B n (z) and C n (z) as linear combinations of the Legendre polynomials are given. ARTICLE HISTORY Received March 07 Accepted June 07 KEYWORDS Legendre functions; parameter derivatives; dilogarithm MSC00 33C05; 33B30. Introduction Over the past 0 years or so, a growth of interest in parameter derivatives of various special functions has been observed. The research done on the subject is documented in a number of papers reporting diverse methods for finding such derivatives for orthogonal polynomials in one [ 3]andtwo[ 6] variables, for Bessel functions [7 0], for Legendre and allied functions [ 8], and also for various types of hypergeometric functions [,9 ]. CONTACT Radosław Szmytkowski radoslaw.szmytkowski@pg.edu.pl Atomicand OpticalPhysics Division, Department of Atomic, Molecular and Optical Physics, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, ul. Gabriela Narutowicza /, Gdańsk, Poland 07 Informa UK Limited, trading as Taylor & Francis Group
2 66 R. SZMYTKOWSKI In Refs. [,], we presented results of our investigations on the first-order derivative of the Legendre function of the first kind with respect to its degree. We showed that [ P ν (z)/] ν=n,withn N 0,isoftheform P ν (z) = P n (z) ln z + + R n (z), (.) where P n (z) is the Legendre polynomial of degree n and R n (z) is another polynomial in z of the same degree. We investigated properties of the polynomials R n (z) and arrived at their several explicit representations, including the following one: R n (z) = [ψ(n + ) ψ(n + )]P n (z) + ( ) n+k k + P k(z), (.) where ψ(z) = dlnɣ(z)/dz is the digamma function. In the year 0, Dr George P. Schramkowski kindly informed the present author that in the course of doing research on a certain problem in theoretical hydrodynamics, he had come across higher-order derivatives [ k P ν (z)/ k ] ν=n,withn N 0 and k. Using Mathematica, Schramkowski found that where = Li ν=0, (.3) Li z = z 0 ln( t) dt t (.) is the dilogaritm function [3,]. In Ref. [5], we gave an analytical proof of the result displayed in Equation (.3), and also we derived a closed-form formula for the third-order derivative [ 3 P ν (z)/ 3 ] ν=0. That work was then extended by Laurenzi [6], who found an expression for the fourth-order derivative [ P ν (z)/ ] ν=0. The primary purpose of the present work is to pursue further the research initiated by Schramkowski and continued by us in Ref. [5]. We shall show that for arbitrary n N 0 the second-order derivative [/ ] ν=n may be expressed in the form = P n (z) Li + B n (z) ln z + + C n (z), (.5) where the polynomials B n (z) and C n (z) have the following representations in terms of the Legendre polynomials: k + B n (z) = [ψ(n + ) ψ(n + )]P n (z) + P k(z) (.6)
3 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 67 and C n (z) = { π } 3 + [ψ(n + ) ψ(n + )] + ψ (n + ) ψ (n + ) P n (z) + ( ) n+k k + ( ) n + k ψ + ( n k + ψ { [ ψ(n + k + ) ψ(n k + ) )] + ( ) n+k k + n + } P k (z), (.7) with ψ(z) being already defined under Equation (.), ψ (z) = dψ(z)/dz, and x = max{n Z : n x} standing for the integer part of x. Infact,theaboveresultfor [/ ] ν=n,validforn N 0, may be easily extended to any n Z, since with the use of the well-known identity one immediately finds that P ν (z) = P ν (z) (.8) = P ν (z) ν= (n N 0 ). (.9) ν=n In addition to the above summarized study on the second-order degree-derivative of the Legendre function of the first kind, which will be presented in detail in Section,later in Section 3 we shall also prove that if z C \ [, ] and n N 0, then the first-order derivative [/] ν=n,whereq ν (z) is the Legendre function of the second kind, is given by = P n (z) Li P n(z) ln z + ln z + B n(z) ln z + ( )n B n ( z) ln z π 6 P n(z) + C n(z) ( )n C n ( z). (.0) A counterpart expression for [ Q ν (x)/] ν=n,applicablewhenx (, ), willalsobe derived.. The derivatives [/ ] ν=n.. The general form of [/ ] ν=n Our point of departure is the well-known recurrence relation (ν + )P ν+ (z) (ν + )zp ν (z) + νp ν (z) = 0 (.)
4 68 R. SZMYTKOWSKI obeyed by the Legendre functions of the first kind. Differentiating it twice with respect to ν and setting then ν = n yields (n + ) P ν (z) (n + )z P ν (z) ν=n+ + n P ν (z) ν=n ν= [ Pν (z) = z P ν(z) ν=n+ + P ] ν(z) ν=n. (.) ν= If we replace the first-order derivatives on the right-hand side with expressions following from Equation (.), this furnishes (n + ) P ν (z) (n + )z P ν (z) ν=n+ + n P ν (z) ν=n = [P n+ (z) zp n (z) + P (z)]ln z + ν= [R n+ (z) zr n (z) + R (z)]. (.3) From the formal point of view, one may look at Equation (.3) as a second-order difference equation and then two additional conditions are necessary to single out the sequence [/ ] ν=n from its general solution. Such conditions may be chosen in a variety of ways but for our purposes it is most convenient to take the explicit expression for [/ ] ν=0, given in Equation (.), as the first one. The second suitable condition follows from Equation (.3) after one lets n = 0. Then, with the use of the identities and [, Sec. 5.] P (z) =, P 0 (z) =, P (z) = z (.) R (z) = ln z +, R 0(z) = 0, R (z) = z, (.5) Equation (.3) reduces to the form z P ν (z) ν= = (z + ) ln z + (z ). (.6) ν=0 On combining Equation (.6) with Equation (.), one finds that = z Li ν= + (z + ) ln z + (z ). (.7) If necessary, Equation (.3) may be applied recursively, with Equations (.3) and (.7) used as initial conditions, to generate the derivative in question for any particular n. However, as we shall show below, it is also possible to obtain a closed-form representation for [/ ] ν=n. As the first step towards that goal, we observe that the structure of Equation (.3), together with explicit expressions for the derivatives [/ ] ν=0 and [/ ] ν=
5 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 69 displayed in Equations (.3) and (.7), fix the form of [/ ] ν=n to be = A n (z) Li + B n (z) ln z + + C n (z), (.8) where A n (z), B n (z) and C n (z) are polynomials in z of degree n. Since the right-hand side of Equation (.3) does not contain the dilogarithm function, the polynomial A n (z) solves the homogeneous recurrence subject to the initial conditions (n + )A n+ (z) (n + )za n (z) + na n (z) = 0 (.9) A 0 (z) = = P 0 (z), A (z) = z = P (z), (.0) which follow from Equations (.) and (.8). Hence, we deduce the following expression for A n (z) in terms of the Legendre polynomial P n (z): Consequently, Equation (.8) becomes = P n (z) Li A n (z) = P n (z). (.) + B n (z) ln z + + C n (z). (.) The representations of the polynomials B n (z) and C n (z) remain to be established... Differential equations for the polynomials B n (z) and C n (z) It is known that the Legendre function P ν (z) obeys the differential identity [ d dz ( ) d ] + ν(ν + ) P ν (z) = 0. (.3) dz If we differentiate it twice with respect to ν and then put ν = n,thisgives [ d dz ( ) d ] + n(n + ) dz = (n + ) P ν(z) ν=n P n (z) (.) ν=n and further, after Equation (.) is plugged into the first term on the right-hand side, [ d dz ( ) d ] + n(n + ) dz = (n + )P n (z) ln z + ν=n (n + )R n (z) P n (z). (.5) Next, we insert Equation (.) into the left-hand side of Equation (.5) and equate those terms appearing on both sides which involve the logarithmic factor. This yields the
6 650 R. SZMYTKOWSKI following inhomogeneous differential equation for B n (z): [ d dz ( ) d ] [ + n(n + ) B n (z) = (z + ) dp ] n(z) np n (z). (.6) dz dz Similarly, after equating polynomial expressions on both sides, we arrive at the inhomogeneousequationforc n (z): [ d dz ( ) d ] + n(n + ) C n (z) = (z ) db n(z) + B n (z) (n + )R n (z). dz dz (.7) Consider Equation (.6). It is evident that it does not possess a unique polynomial solution, since to any particular solution of that form one may add an arbitrary multiple of the Legendre polynomial P n (z), which results in another polynomial solution. To make the polynomial solution unique, we thus need an additional constraint. The latter follows from the limiting relation [7] P ν (z) z sin(πν) ln z + + O(), (.8) π from which we find ν=n z O(). (.9) Hence, the left-hand side of Equation (.) remains finite for z andtomakethe right-hand side also finite in that limit, we are forced to put B n ( ) = 0. (.0) If, in turn, we wish to make the polynomial solution to Equation (.7) unique, we use the identity Differentiating twice with respect to ν,weobtain P ν () =. (.) P ν () = 0. (.) ν=n Since Li 0 = 0andln= 0, we deduce that C n (z) is constrained to obey C n () = 0. (.3) Below we shall exploit Equations (.6), (.0), (.7) and (.3) to determine the polynomials B n (z) and C n (z).
7 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS Construction of the polynomials B n (z) A general form of the polynomials B n (z) maybeobtainedwithease.inref.[, Sec. 5..], we have found that the polynomials R n (z) obey the differential relation [ d dz ( ) d ] [ + n(n + ) R n (z) = (z ) dp ] n(z) np n (z). (.) dz dz Hence, with the use of the well-known identity P n ( z) = ( ) n P n (z), (.5) we deduce that [ d dz ( ) d ] [ + n(n + ) R n ( z) = ( ) n (z + ) dp ] n(z) np n (z). (.6) dz dz Comparison of Equations (.6) and (.6) shows that the polynomial B n (z) must be of the form B n (z) = ( ) n R n ( z) + b n P n (z). (.7) To determine the constant b n,weputz = in Equation (.7). By virtue of the constraint (.0), with the use of the relations and(cf. Ref.[, Equation (5.0)]) we infer that and thus finally we arrive at P n ( ) = ( ) n (.8) R n () = 0, (.9) b n = 0, (.30) B n (z) = ( ) n R n ( z). (.3) On combining Equation (.3) with Equation (.), we have the following explicit representation of B n (z): k + B n (z) = [ψ(n + ) ψ(n + )]P n (z) + P k(z). (.3) Further expressions for B n (z) may be obtained if one combines Equation (.3) with Equations (.), (.), (.) and (5.90) from Ref. [] or with Equations () and () from Ref. [].
8 65 R. SZMYTKOWSKI.. Construction of the polynomials C n (z) We shall seek a representation of C n (z) in the form of a linear combination of Legendre polynomials: C n (z) = n c nk P k (z). (.33) Action on both sides of Equation (.33) with the Legendre differential operator appearing on the left-hand side of Equation (.7) gives [ d dz ( ) d ] + n(n + ) C n (z) = c nk P k (z). (.3) dz On the other side, with the aid of Equations (.) and (.3), and of the identity (z ) dp n(z) dz = np n (z) + ( ) n+k (k + )P k (z), (.35) after some algebra we find that the expression on the right-hand side of Equation (.7) may be written as (z ) db n(z) + B n (z) (n + )R n (z) dz { = ( ) n+k 8(k + )[ψ(n + ) ψ(n + )] + 8(k + ) ( ) m+k m + (n m)(n + m + ) m=k (k + ) Equations (.7), (.3) and (.36) yield ( )n+k (n + )(k + ) c nk = ( ) n+k 8(k + ) [ψ(n + ) ψ(n + )] 8(k + ) + ( ) m+k m + (n m)(n + m + ) m=k } P k (z). (.36) (k + ) (n k) (n + k + ) ( )n+k (n + )(k + ) (n k) (n + k + ) (0 k n ). (.37)
9 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 653 It is proven in Appendix A. that ( ) n+m m + (n m)(n + m + ) m=k = ψ(n + ) + ψ(n + k + ) + ψ(n + ) ψ(n k + ) ( ) ( ) n + k n k ψ + + ψ + (0 k n ), (.38) where x =max{n Z : n x}. Use of Equation (.38) casts Equation (.37) into the final form c nk = ( ) n+k 8(k + ) [ ( ) n + k ψ(n + k + ) ψ(n k + ) ψ + + ψ ( n k )] + (k + ) (n k) (n + k + ) ( )n+k (n + )(k + ) (n k) (n + k + ) (0 k n ). (.39) Equation (.39) says nothing about the coefficient c nn. But from Equations (.3), (.) and (.33) it can be deduced that c nn may be expressed as c nn = c nk. (.0) This implies that the polynomial C n (z) may be written as C n (z) = c nk [P k (z) P n (z)], (.) or explicitly, if the result in Equation (.39) is used, as C n (z) = ( ) n+k k + { [ ψ(n + k + ) ψ(n k + ) ψ ( n k + ψ n + ( n + k )] + ( ) n+k k + ) + } [P k (z) P n (z)]. (.)
10 65 R. SZMYTKOWSKI AbitdifferentformulaforC n (z) is obtained if the coefficient c nn is expressed in a closed form. To find the latter, we combine Equations (.37) and (.0) and write c nn = 8[ψ(n + ) ψ(n + )] ( ) n+k k + 8 ( ) k k + ( ) m m + (n m)(n + m + ) + m=k (k + ) (n k) (n + k + ) + ( ) n+k (n + )(k + ) (n k) (n + k + ). (.3) The sums appearing in Equation (.3) are evaluated in individual subsections of the appendix, where it is found that ( ) n+k k + = ψ(n + ) + ψ(n + ), (.) ( ) k k + ( ) m m + (n m)(n + m + ) m=k = π γ n + + [ψ(n + ) ψ(n + )] n + ψ(n + ) ψ (n + ), (.5) and (k + ) (n k) (n + k + ) = π 6 γ n + n + ψ(n + ) ψ (n + ) (.6) ( ) n+k (n + )(k + ) (n k) (n + k + ) = π + ψ (n + ) ψ (n + ), (.7) with γ standing for the Euler Mascheroni constant and with ψ (ζ ) = dψ(ζ)/dζ being the trigamma function. Plugging the results (.) (.7) into the right-hand side of Equation (.3) furnishes the coefficient c nn in the compact form c nn = π 3 + [ψ(n + ) ψ(n + )] + ψ (n + ) ψ (n + ). (.8)
11 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 655 Hence, by virtue of Equations (.33), (.39) and (.8), we eventually arrive at the sought formula C n (z) = { π } 3 + [ψ(n + ) ψ(n + )] + ψ (n + ) ψ (n + ) P n (z) { [ + ( ) n+k k + ψ(n + k + ) ψ(n k + ) ( ) n + k ψ + ( n k + ψ )] + ( ) n+k k + n + } P k (z), (.9) alternative to the one in Equation (.)..5. Explicit expressions for [/ ] ν=n with 0 n 3 Itmaybeofinteresttoseehowthederivatives[/ ] ν=n look explicitly for several lowest values of n. From Equations (.), (.3) and (.9), for 0 n 3wefindthat = Li ν=0, (.50a) = z Li + (z + ) ln z + z +, (.50b) ν= ( = ( 3z 7 + ) Li + ν= z + 3z ) ln z + z + 5 z +, (.50c) ( = ( 5z z) Li + ν=3 6 z3 + 5z 5 z ) ln z z z + 9 z 0 9. (.50d) 3. The derivatives [/] ν=n and [ Q ν (x)/] ν=n In Refs. [,], we exploited representations of the first-order derivatives [ P ν (z)/] ν=n found therein to obtain expressions for the Legendre functions of the second kind Q n (z), with n N 0,bothforz C \ [, ] and for z = x (, ). Below we shall show that the knowledge of the second-order derivatives [/ ] ν=n allows one to obtain explicit formulas for the first-order derivatives [/] ν=n and [ Q ν (x)/] ν=n,againwith n N 0.
12 656 R. SZMYTKOWSKI 3.. The derivatives [/] ν=n for z C \ [, ] 3... The general form of [/] ν=n The Legendre function of the second kind, Q ν (z), may be defined in terms of the function ofthefirstkindthroughtheformula Q ν (z) = π e iπν P ν (z) P ν ( z) sin(π ν) (Im z 0). (3.) Hence, it follows that π = sin (πν) + { π[p ν (z) P ν ( z) cos(πν)] [ e iπν P ν(z) P ν( z) ] } sin(π ν) From this, for ν = n N 0, with the use of the L Hospital rule, we obtain (Im z 0). (3.) = π ν=n P n(z) iπ P ν (z) + ν=n ν=n ( )n P ν ( z) (Im z 0). (3.3) ν=n If in the above formula the second-order derivatives [ P ν (±z)/ ] ν=n are substituted with expressions following from Equation (.) and the first-order derivative [ P ν (z)/] ν=n is replaced by the right-hand side of Equation (.), this yields [/] ν=n in the form = P n(z) ( ) [ z + Li Li + B n(z) iπ ] P n(z) ln z + ( )n B n ( z) ln π P n(z) iπ R n(z) + C n(z) ( )n C n ( z) (Im z 0). (3.) A more elegant expression for [/] ν=n follows if the dilogarithm Li [(z + )/] is eliminated from Equation (3.) with the aid of the Euler s identity [3, Equation (.)] Li z + Li () = π 6 ln z ln(), (3.5) the relation = e iπ (z ) (Im z 0) (3.6)
13 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 657 and the result in Equation (.3). Proceeding in that way, one eventually finds that = P n (z) Li P n(z) ln z + ln z ( )n B n ( z) ln z π 6 P n(z) + C n(z) + B n(z) ln z + ( )n C n ( z) (n N 0 ). (3.7) 3... Explicit expressions for [/] ν=n with 0 n 3 Explicit forms of the derivatives [/] ν=n with 0 n 3, obtained from Equation (3.7) with the use of Equations (.3) and (.9), are = Li ν=0 ln z + = zli ν= z ln z + ( ) + ln z + + = ν= z + ( 3 z + ( z + 3 z 8 ) Li ln z ln z ( z + π 6, (3.8a) ) ln z ( + 3 z + ) ln z + ) ln z + + ( 7 8 z + 3 z + 8 π z +, (3.8b) 6 ln z ) ln z π z + 5 z + π, (3.8c) = ( 5 z3 + 3 ) z Li + ( 5 ν=3 z3 + 3 ) z ln z + ln z ( ) 37 + ln z + + z3 + 5 z 5 8 z 3 ( 37 z3 + 5 z z 3 ) ln z 5π z3 + 3 z + π z 5 9. (3.8d) Wefinditremarkablethatcoefficientsinthepolynomialpartof[/] ν=n are alternately irrational and rational. 3.. The derivatives [ Q ν (x)/] ν=n for < x < On the real interval < x <, the Legendre function of the second kind, Q ν (x), is defined as the average of the limits Q ν (x + i0) and Q ν (x i0) resulting when z approaches
14 658 R. SZMYTKOWSKI x from the upper (Im z > 0) and lower (Im z < 0) half-planes, respectively. One has Q ν (x) = [Q ν(x + i0) + Q ν (x i0)] ( < x < ), (3.9) and consequently Q ν (x) = Q ν (x + i0) + From this, with the use of Equation (3.7) and the identity Q ν (x i0). (3.0) ν=n x ± i0 = e ±iπ ( x) ( < x < ), (3.) one finds that Q ν (x) x = P n (x) Li + B n(x) ln + x P n(x) ln + x ln x ( )n B n ( x) ln x π 6 P n(x) + C n(x) ( )n C n ( x) (n N 0 ). (3.) There is no need to provide here explicit representations for the derivatives [ Q ν (x)/] ν=n for several lowest non-negative values of n. AsitisseenfromEquations (3.7) and (3.), such representations for 0 n 3 may be immediately deduced from Equations (3.8a) (3.8d) after the replacement of z with x is made everywhere in the latter set of equations, except for the logarithm ln[(z )/], which is to be substituted with ln[( x)/]. Acknowledgments I wish to thank Dr George P. Schramkowski for kindly communicating to me the formula in Equation (.3) and for the subsequent inspiring correspondence. Disclosure statement No potential conflict of interest was reported by the author. References [] Froehlich J. Parameter derivatives of the Jacobi polynomials and the Gaussian hypergeometric function. Integral Transforms Spec Funct. 99;: [] Koepf W. Identities for families of orthogonal polynomials and special functions. Integral Transforms Spec Funct. 997;5:69 0. [3] Szmytkowski R. A note on parameter derivatives of classical orthogonal polynomials; 009. arxiv:
15 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 659 [] AktaşR.AnoteonparameterderivativesoftheJacobipolynomialsonthetriangle.ApplMath Comput. 0;7: [5] Aktaş R. On parameter derivatives of a family of polynomials in two variables. Appl Math Comput. 05;56: [6] Aktaş R. Representations for parameter derivatives of some Koornwinder polynomials in two variables. J Egyptian Math Soc. 06;: [7] Brychkov YuA, Geddes KO. On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec Funct. 005;6: [8] Sesma J. Derivatives with respect to the order of the Bessel function of the first kind; 0. arxiv: [9] Dunster TM. On the order derivatives of Bessel functions. Constr Approx. 06. doi:0.007/s [0] Brychkov YuA. Higher derivatives of the Bessel functions with respect to the order. Integral Transforms Spec Funct. 06;7: [] Szmytkowski R. On the derivative of the Legendre function of the first kind with respect to its degree. J Phys A. 006;39:57 57 [Corrigendum: J Phys A. 007;0: ]. [] Szmytkowski R. Addendum to On the derivative of the Legendre function of the first kind with respect to its degree. J Phys A. 007;0: [3] Szmytkowski R. On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J Math Chem. 009;6:3 60. [] Brychkov YuA. On the derivatives of the Legendre functions P μ ν (z) and Q μ ν (z) with respect to μ and ν. Integral Transforms Spec Funct. 00;:75 8. [5] Cohl HS. Derivatives with respect to the degree and order of associated Legendre functions for z > using modified Bessel functions. Integral Transforms Spec Funct. 00;: [6] Cohl HS. On parameter differentiation for integral representations of associated Legendre functions. Sym Integ Geom: Meth Appl (SIGMA). 0;7:050. [7] Szmytkowski R. On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J Math Chem. 0;9: [8] Szmytkowski R. On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J Math Anal Appl. 0;386:33 3. [9] Abad J, Sesma J. Successive derivatives of Whittaker functions with respect to the first parameter. Comput Phys Commun. 003;56:3. [0] Ancarani LU, Gasaneo G. Derivatives of any order of the confluent hypergeometric function F (a, b, z) with respect to the parameter a or b.jmathphys.008;9: [] Ancarani LU, Gasaneo G. Derivatives of any order of the Gaussian hypergeometric function F (a, b, c; z) with respect to the parameters a, b and c.jphysa.009;: [] Ancarani LU, Gasaneo G. Derivatives of any order of the hypergeometric function pf q (a,..., a p ; b,..., b q ; z) with respect to the parameters a i and b i.jphysa.00;3: [3] Lewin L. Polylogarithms and associated functions. New York (NY): North-Holland; 98. [] Apostol TM. Zeta and related functions. In: Olver FWJ, Lozier DW, Boisvert RF, Clark CW, editors. NIST handbook of mathematical functions. Cambridge (UK): Cambridge University Press; 00. Section 5.. [5] Szmytkowski R. The derivatives [/ ] ν=0 and [ 3 P ν (z)/ 3 ] ν=0,wherep ν (z) is the Legendre function of the first kind; 03. arxiv: [6] Laurenzi BJ. Derivatives with respect to the order of the Legendre polynomials; 05. arxiv: [7] Magnus W, Oberhettinger F, Soni RP. Formulas and theorems for the special functions of mathematical physics. 3rd ed. Berlin: Springer; 966.
16 660 R. SZMYTKOWSKI Appendix. Proofs of summation formulas used in Section. A. The summation formulas (.38) and (.) We denote We have and further S = n m k= S = ( ) n+k k + k=m ( ) k + k However, it is easy to show that S = n k=m k=n+m+ ( ) n+k n k ( ) k = k n m k= k=m (0 m n ). (A) ( ) n+k n + k + ( ) k + k N ( ) k N N/ = k k + k k= k= k= n k= ( ) k where x =max{n Z : n x} stands for the integer part of x.since N k= k n+m k= (A) ( ) k. (A3) k (N N 0 ), (A) k = ψ(n + ) ψ() (N N 0), (A5) with ψ(z) = dlnɣ(z)/dz being the digamma function, Equation (A) may be rewritten in the form N ( ) k ( ) N = ψ(n + ) + ψ + (N N 0 ). (A6) k k= Application of the result (A6) to each of the three sums on the extreme right-hand side of Equation (A3) gives finally ( ) n+k k + k=m = ψ(n + ) + ψ(n + m + ) + ψ(n + ) ψ(n m + ) ( ) ( ) n + m n m ψ + + ψ + (0 m n ). (A7) After k is interchanged with m, Equation (A7) becomes identical with Equation (.38). For m = 0, Equation (A7) becomes ( ) n+k k + = ψ(n + ) + ψ(n + ), (A8) which is Equation (.). A. The summation formula (.5) We denote S = ( ) k k + ( ) m m + (n m)(n + m + ). (A9) m=k
17 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 66 Application of the identity N f k k=n m=k transforms Equation (A9) into S = [ N f m = N k=n f k ( ) k k + + N fk (N N ) (A0) k=n ] + from which, with the help of Equations (A8) and (A), we obtain ( ) k k + ( ) m m + (n m)(n + m + ) m=k (k + ) (n k) (n + k + ), (A) = π γ n + + [ψ(n + ) ψ(n + )] n + ψ(n + ) ψ (n + ), (A) which is Equation (.5). Toprovetheidentity(A0),wewritetheobviouschainofequalities(N N is assumed) N k=n f k = N f k k=n N m=n f m = N f k k=n k f m + m=n N N f k k=n m=k N f m fk. k=n Manipulating with the first term on the extreme right-hand side of Equation (A3), we have (A3) N f k k=n k f m = m=n N f m m=n N f k = k=m N N f k k=n m=k Plugging the result (A) into Equation (A3), we obtain N N N N f k = f k f m fk, k=n k=n m=k k=n from which the identity in Equation (A0) follows immediately. f m. (A) (A5) A.3 The summation formula (.6) We denote S 3 = (k + ) (n k) (n + k + ). (A6) If we carry out the partial fraction decomposition of the summand, we have S 3 = (n k) + and further, after obvious rearrangements, (n + k + ) n + S 3 = (k + ) n + n k n + n k= k. n + k + (A7) (A8)
18 66 R. SZMYTKOWSKI Now, it holds that N (k + ) = (k + ) (k + N + ) = ψ () ψ (N + ) (N N 0 ), (A9) where ψ (z) = dψ(z)/dz is the trigamma function. On employing Equations (A9) and (A5) in Equation (A8), after using the well-known relations ψ() = γ (here and below γ stands for the Euler Mascheroni constant) and we finally obtain ψ () = π (A0) 6, (A) (k + ) (n k) (n + k + ) = π 6 γ n + n + ψ(n + ) ψ (n + ), which is Equation (.6). (A) A. The summation formula (.7) We denote S = ( ) n+k (n + )(k + ) (n k) (n + k + ). (A3) A partial-fraction decomposition of the summand gives S = ( ) n+k (n k) ( ) n+k (n + k + ). (A) Withalittlebitofalgebraontheright-handsideofEquation(A),weobtain S = ( ) k+ (k + ) (A5) and further S = (k + ) + (k + ). (A6) From this, with reference to Equations (A9) and (A), we eventually arrive at ( ) n+k (n + )(k + ) (n k) (n + k + ) = π + ψ (n + ) ψ (n + ), (A7) which is Equation (.7).
Addendum to On the derivative of the Legendre function of the first kind with respect to its degree
IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (007) 14887 14891 doi:10.1088/1751-8113/40/49/00 ADDENDUM Addendum to On the derivative of the Legendre function
More informationarxiv: v3 [math.ca] 16 Nov 2010
A note on parameter derivatives of classical orthogonal polynomials Rados law Szmytowsi arxiv:0901.639v3 [math.ca] 16 Nov 010 Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty
More informationSome integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut ( 1, 1)
Integral Transforms and Special Functions Vol. 3, No. 11, November 01, 847 85 Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut, 1) Radosław Szmytkowski*
More informationIntegral Transforms and Special Functions, 2014 Vol. 25, No. 4, ,
Integral Transforms and Special Functions, 014 Vol. 5, No. 4, 31 317, http://dx.doi.org/10.1080/1065469.013.8435 A Dirac delta-type orthogonality relation for the on-the-cut generalized associated Legendre
More informationFirst published on: 10 December 2009 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [University of Auckland] On: August 1 Access details: Access Details: [subscription number 97449359] Publisher Taylor & Francis Informa Ltd Registered in England and Wales
More informationSeries solutions to a second order linear differential equation with regular singular points
Physics 6C Fall 0 Series solutions to a second order linear differential equation with regular singular points Consider the second-order linear differential equation, d y dx + p(x) dy x dx + q(x) y = 0,
More informationNew series expansions of the Gauss hypergeometric function
New series expansions of the Gauss hypergeometric function José L. López and Nico. M. Temme 2 Departamento de Matemática e Informática, Universidad Pública de Navarra, 36-Pamplona, Spain. e-mail: jl.lopez@unavarra.es.
More informationarxiv: v6 [math.nt] 12 Sep 2017
Counterexamples to the conjectured transcendence of /α) k, its closed-form summation and extensions to polygamma functions and zeta series arxiv:09.44v6 [math.nt] Sep 07 F. M. S. Lima Institute of Physics,
More informationarxiv: v2 [math.ca] 2 Sep 2017
A note on the asymptotics of the modified Bessel functions on the Stokes lines arxiv:1708.09656v2 [math.ca] 2 Sep 2017 R. B. Paris Division of Computing and Mathematics, University of Abertay Dundee, Dundee
More informationON SIMILARITIES BETWEEN EXPONENTIAL POLYNOMIALS AND HERMITE POLYNOMIALS
Journal of Applied Mathematics and Computational Mechanics 2013, 12(3), 93-104 ON SIMILARITIES BETWEEN EXPONENTIAL POLYNOMIALS AND HERMITE POLYNOMIALS Edyta Hetmaniok, Mariusz Pleszczyński, Damian Słota,
More informationarxiv: v2 [math.ca] 19 Oct 2012
Symmetry, Integrability and Geometry: Methods and Applications Def inite Integrals using Orthogonality and Integral Transforms SIGMA 8 (, 77, pages Howard S. COHL and Hans VOLKMER arxiv:.4v [math.ca] 9
More informationConsidering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.
Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and
More informationBeukers integrals and Apéry s recurrences
2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (25), Article 5.. Beukers integrals and Apéry s recurrences Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G CANADA lkjain@uwaterloo.ca
More informationAppendix A Vector Analysis
Appendix A Vector Analysis A.1 Orthogonal Coordinate Systems A.1.1 Cartesian (Rectangular Coordinate System The unit vectors are denoted by x, ŷ, ẑ in the Cartesian system. By convention, ( x, ŷ, ẑ triplet
More informationCompleteness of the Dirac oscillator eigenfunctions
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 34 (001) 4991 4997 www.iop.org/journals/ja PII: S0305-4470(01)011-X Completeness of the Dirac oscillator
More informationThe Expansion of the Confluent Hypergeometric Function on the Positive Real Axis
Applied Mathematical Sciences, Vol. 12, 2018, no. 1, 19-26 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712351 The Expansion of the Confluent Hypergeometric Function on the Positive Real
More informationClosed-form Second Solution to the Confluent Hypergeometric Difference Equation in the Degenerate Case
International Journal of Difference Equations ISS 973-669, Volume 11, umber 2, pp. 23 214 (216) http://campus.mst.edu/ijde Closed-form Second Solution to the Confluent Hypergeometric Difference Equation
More informationCorrelated exponential functions in high precision calculations for diatomic molecules. Abstract
Version 3.1 Correlated exponential functions in high precision calculations for diatomic molecules Krzysztof Pachucki Faculty of Physics University of Warsaw Hoża 69 00-681 Warsaw Poland Abstract Various
More informationOn A Central Binomial Series Related to ζ(4).
On A Central Binomial Series Related to ζ. Vivek Kaushik November 8 arxiv:8.67v [math.ca] 5 Nov 8 Abstract n n n 7π 3 by We prove a classical binomial coefficient series identity n evaluating a double
More informationCertain inequalities involving the k-struve function
Nisar et al. Journal of Inequalities and Applications 7) 7:7 DOI.86/s366-7-343- R E S E A R C H Open Access Certain inequalities involving the -Struve function Kottaaran Sooppy Nisar, Saiful Rahman Mondal
More informationAsymptotics of Integrals of. Hermite Polynomials
Applied Mathematical Sciences, Vol. 4, 010, no. 61, 04-056 Asymptotics of Integrals of Hermite Polynomials R. B. Paris Division of Complex Systems University of Abertay Dundee Dundee DD1 1HG, UK R.Paris@abertay.ac.uk
More informationOn Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable
Communications in Mathematics and Applications Volume (0), Numbers -3, pp. 97 09 RGN Publications http://www.rgnpublications.com On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable A. Shehata
More informationSeries solutions of second order linear differential equations
Series solutions of second order linear differential equations We start with Definition 1. A function f of a complex variable z is called analytic at z = z 0 if there exists a convergent Taylor series
More informationVALUES OF THE LEGENDRE CHI AND HURWITZ ZETA FUNCTIONS AT RATIONAL ARGUMENTS
MATHEMATICS OF COMPUTATION Volume 68, Number 228, Pages 1623 1630 S 0025-5718(99)01091-1 Article electronically published on May 17, 1999 VALUES OF THE LEGENDRE CHI AND HURWITZ ZETA FUNCTIONS AT RATIONAL
More informationON VALUES OF THE PSI FUNCTION
Journal of Applied Mathematics and Computational Mechanics 07, 6(), 7-8 www.amcm.pcz.pl p-issn 99-9965 DOI: 0.75/jamcm.07..0 e-issn 353-0588 ON VALUES OF THE PSI FUNCTION Marcin Adam, Bożena Piątek, Mariusz
More informationNonlinear Integral Equation Formulation of Orthogonal Polynomials
Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli Ben-Naim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. Ben-Naim,
More informationrama.tex; 21/03/2011; 0:37; p.1
rama.tex; /03/0; 0:37; p. Multiple Gamma Function and Its Application to Computation of Series and Products V. S. Adamchik Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA Abstract.
More informationOn the de Branges and Weinstein Functions
On the de Branges and Weinstein Functions Prof. Dr. Wolfram Koepf University of Kassel koepf@mathematik.uni-kassel.de http://www.mathematik.uni-kassel.de/~koepf Tag der Funktionentheorie 5. Juni 2004 Universität
More informationPolyexponentials. Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH
Polyexponentials Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH 45810 k-boyadzhiev@onu.edu 1. Introduction. The polylogarithmic function [15] (1.1) and the more general
More informationTransformation formulas for the generalized hypergeometric function with integral parameter differences
Transformation formulas for the generalized hypergeometric function with integral parameter differences A. R. Miller Formerly Professor of Mathematics at George Washington University, 66 8th Street NW,
More information5.4 Bessel s Equation. Bessel Functions
SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent
More informationJournal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics 233 2010) 1554 1561 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: wwwelseviercom/locate/cam
More informationOpen problems. Christian Berg a a Department of Mathematical Sciences, University of. Copenhagen, Copenhagen, Denmark Published online: 07 Nov 2014.
This article was downloaded by: [Copenhagen University Library] On: 4 November 24, At: :7 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 72954 Registered office:
More informationQuadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers
Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers Martin Nicholson In this brief note, we show how to apply Kummer s and other quadratic transformation formulas for
More informationContents. I Basic Methods 13
Preface xiii 1 Introduction 1 I Basic Methods 13 2 Convergent and Divergent Series 15 2.1 Introduction... 15 2.1.1 Power series: First steps... 15 2.1.2 Further practical aspects... 17 2.2 Differential
More informationRelevant sections from AMATH 351 Course Notes (Wainwright): Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls):
Lecture 5 Series solutions to DEs Relevant sections from AMATH 35 Course Notes (Wainwright):.4. Relevant sections from AMATH 35 Course Notes (Poulin and Ingalls): 2.-2.3 As mentioned earlier in this course,
More informationNormalization integrals of orthogonal Heun functions
Normalization integrals of orthogonal Heun functions Peter A. Becker a) Center for Earth Observing and Space Research, Institute for Computational Sciences and Informatics, and Department of Physics and
More informationBilinear generating relations for a family of q-polynomials and generalized basic hypergeometric functions
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 16, Number 2, 2012 Available online at www.math.ut.ee/acta/ Bilinear generating relations for a family of -polynomials and generalized
More informationB n (x) zn n! n=0. E n (x) zn n! n=0
UDC 517.9 Q.-M. Luo Chongqing Normal Univ., China) q-apostol EULER POLYNOMIALS AND q-alternating SUMS* q-полiноми АПОСТОЛА ЕЙЛЕРА ТА q-знакозмiннi СУМИ We establish the basic properties generating functions
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationON A NEW CLASS OF INTEGRALS INVOLVING PRODUCT OF GENERALIZED BESSEL FUNCTION OF THE FIRST KIND AND GENERAL CLASS OF POLYNOMIALS
Acta Universitatis Apulensis ISSN: 158-59 http://www.uab.ro/auajournal/ No. 6/16 pp. 97-15 doi: 1.1711/j.aua.16.6.8 ON A NEW CLASS OF INTEGRALS INVOLVING PRODUCT OF GENERALIZED BESSEL FUNCTION OF THE FIRST
More informationCONTINUED-FRACTION EXPANSIONS FOR THE RIEMANN ZETA FUNCTION AND POLYLOGARITHMS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 9, September 997, Pages 2543 2550 S 0002-9939(97)0402-6 CONTINUED-FRACTION EXPANSIONS FOR THE RIEMANN ZETA FUNCTION AND POLYLOGARITHMS
More informationCOMPUTATION OF BESSEL AND AIRY FUNCTIONS AND OF RELATED GAUSSIAN QUADRATURE FORMULAE
BIT 6-85//41-11 $16., Vol. 4, No. 1, pp. 11 118 c Swets & Zeitlinger COMPUTATION OF BESSEL AND AIRY FUNCTIONS AND OF RELATED GAUSSIAN QUADRATURE FORMULAE WALTER GAUTSCHI Department of Computer Sciences,
More informationVector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 032, pages Vector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials Helene AIRAULT LAMFA
More informationThe Asymptotic Expansion of a Generalised Mathieu Series
Applied Mathematical Sciences, Vol. 7, 013, no. 15, 609-616 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.013.3949 The Asymptotic Expansion of a Generalised Mathieu Series R. B. Paris School
More informationCalculation of Cylindrical Functions using Correction of Asymptotic Expansions
Universal Journal of Applied Mathematics & Computation (4), 5-46 www.papersciences.com Calculation of Cylindrical Functions using Correction of Asymptotic Epansions G.B. Muravskii Faculty of Civil and
More informationHigher-order ordinary differential equations
Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationOn Turán s inequality for Legendre polynomials
Expo. Math. 25 (2007) 181 186 www.elsevier.de/exmath On Turán s inequality for Legendre polynomials Horst Alzer a, Stefan Gerhold b, Manuel Kauers c,, Alexandru Lupaş d a Morsbacher Str. 10, 51545 Waldbröl,
More informationInfinite Series. 1 Introduction. 2 General discussion on convergence
Infinite Series 1 Introduction I will only cover a few topics in this lecture, choosing to discuss those which I have used over the years. The text covers substantially more material and is available for
More informationWavelets and polylogarithms of negative integer order
Wavelets and polylogarithms of negative integer order T.E.Krenkel, E.T.Krenkel, K.O.Egiazarian, and J.T.Astola Moscow Technical University for Communications and Informatics, Department of Computer Science,
More informationAn Analysis of Five Numerical Methods for Approximating Certain Hypergeometric Functions in Domains within their Radii of Convergence
An Analysis of Five Numerical Methods for Approximating Certain Hypergeometric Functions in Domains within their Radii of Convergence John Pearson MSc Special Topic Abstract Numerical approximations of
More informationDifferential Equations and Associators for Periods
Differential Equations and Associators for Periods Stephan Stieberger, MPP München Workshop on Geometry and Physics in memoriam of Ioannis Bakas November 2-25, 26 Schloß Ringberg, Tegernsee based on: St.St.,
More informationCONGRUENCES FOR BERNOULLI - LUCAS SUMS
CONGRUENCES FOR BERNOULLI - LUCAS SUMS PAUL THOMAS YOUNG Abstract. We give strong congruences for sums of the form N BnVn+1 where Bn denotes the Bernoulli number and V n denotes a Lucas sequence of the
More informationarxiv: v2 [math.nt] 2 Jul 2013
SECANT ZETA FUNCTIONS MATILDE LALÍN, FRANCIS RODRIGUE, AND MATHEW ROGERS arxiv:304.3922v2 [math.nt] 2 Jul 203 Abstract. Westudytheseries ψ sz := secnπzn s, andprovethatit converges under mild restrictions
More informationOn Recurrences for Ising Integrals
On Recurrences for Ising Integrals Flavia Stan Research Institute for Symbolic Computation (RISC-Linz) Johannes Kepler University Linz, Austria December 7, 007 Abstract We use WZ-summation methods to compute
More informationBessel Functions Michael Taylor. Lecture Notes for Math 524
Bessel Functions Michael Taylor Lecture Notes for Math 54 Contents 1. Introduction. Conversion to first order systems 3. The Bessel functions J ν 4. The Bessel functions Y ν 5. Relations between J ν and
More informationHYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES
HYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES ABDUL HASSEN AND HIEU D. NGUYEN Abstract. There are two analytic approaches to Bernoulli polynomials B n(x): either by way of the generating function
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS
Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific (2007 ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS ANASTASIOS
More informationDifference Equations for Multiple Charlier and Meixner Polynomials 1
Difference Equations for Multiple Charlier and Meixner Polynomials 1 WALTER VAN ASSCHE Department of Mathematics Katholieke Universiteit Leuven B-3001 Leuven, Belgium E-mail: walter@wis.kuleuven.ac.be
More informationarxiv:math/ v1 [math.ca] 21 Mar 2006
arxiv:math/0603516v1 [math.ca] 1 Mar 006 THE FOURTH-ORDER TYPE LINEAR ORDINARY DIFFERENTIAL EQUATIONS W.N. EVERITT, D. J. SMITH, AND M. VAN HOEIJ Abstract. This note reports on the recent advancements
More informationPolyGamma Functions of Negative Order
Carnegie Mellon University Research Showcase @ CMU Computer Science Department School of Computer Science -998 PolyGamma Functions of Negative Order Victor S. Adamchik Carnegie Mellon University Follow
More informationON GENERATING FUNCTIONS OF THE JACOBI POLYNOMIALS
ON GENERATING FUNCTIONS OF THE JACOBI POLYNOMIALS PETER HENRICI 1. Introduction. The series of Jacobi polynomials w = 0 (a n independent of p and τ) has in the case a n =l already been evaluated by Jacobi
More informationON GENERALIZED WEYL FRACTIONAL q-integral OPERATOR INVOLVING GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS. Abstract
ON GENERALIZED WEYL FRACTIONAL q-integral OPERATOR INVOLVING GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS R.K. Yadav 1, S.D. Purohit, S.L. Kalla 3 Abstract Fractional q-integral operators of generalized
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationMATRIX REPRESENTATIONS FOR MULTIPLICATIVE NESTED SUMS. 1. Introduction. The harmonic sums, defined by [BK99, eq. 4, p. 1] sign (i 1 ) n 1 (N) :=
MATRIX REPRESENTATIONS FOR MULTIPLICATIVE NESTED SUMS LIN JIU AND DIANE YAHUI SHI* Abstract We study the multiplicative nested sums which are generalizations of the harmonic sums and provide a calculation
More informationThe Gauss hypergeometric function F (a, b; c; z) for large c
The Gauss hypergeometric function F a, b; c; z) for large c Chelo Ferreira, José L. López 2 and Ester Pérez Sinusía 2 Departamento de Matemática Aplicada Universidad de Zaragoza, 5003-Zaragoza, Spain.
More informationIntegrals evaluated in terms of Catalan s constant. Graham Jameson and Nick Lord (Math. Gazette, March 2017)
Integrals evaluated in terms of Catalan s constant Graham Jameson and Nick Lord (Math. Gazette, March 7) Catalan s constant, named after E. C. Catalan (84 894) and usually denoted by G, is defined by G
More informationThe Derivative Of A Finite Continued Fraction. Jussi Malila. Received 4 May Abstract
Applied Mathematics E-otes, 142014, 13-19 c ISS 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ The Derivative Of A Finite Continued Fraction Jussi Malila Received 4 May
More informationAnalogues for Bessel Functions of the Christoffel-Darboux Identity
Analogues for Bessel Functions of the Christoffel-Darboux Identity Mark Tygert Research Report YALEU/DCS/RR-1351 March 30, 2006 Abstract We derive analogues for Bessel functions of what is known as the
More informationNew asymptotic expansion for the Γ (z) function.
New asymptotic expansion for the Γ z function. Gergő Nemes Institute of Mathematics, Eötvös Loránd University 7 Budapest, Hungary September 4, 007 Published in Stan s Library, Volume II, 3 Dec 007. Link:
More informationThe evaluation of integrals of Bessel functions via G-function identities
The evaluation of integrals of Bessel functions via G-function identities Victor Adamchik Wolfram earch Inc., 1 Trade Center Dr., Champaign, IL 6182, USA Abstract A few transformations are presented for
More informationResearch Article Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 80515, 9 pages doi:10.1155/2007/80515 Research Article Operator Representation of Fermi-Dirac
More informationarxiv: v1 [hep-ph] 30 Dec 2015
June 3, 8 Derivation of functional equations for Feynman integrals from algebraic relations arxiv:5.94v [hep-ph] 3 Dec 5 O.V. Tarasov II. Institut für Theoretische Physik, Universität Hamburg, Luruper
More informationProducts of random matrices
PHYSICAL REVIEW E 66, 664 Products of random matrices A. D. Jackson* and B. Lautrup The Niels Bohr Institute, Copenhagen, Denmark P. Johansen The Institute of Computer Science, University of Copenhagen,
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationParametric Euler Sum Identities
Parametric Euler Sum Identities David Borwein, Jonathan M. Borwein, and David M. Bradley September 23, 2004 Introduction A somewhat unlikely-looking identity is n n nn x m m x n n 2 n x, valid for all
More informationMath 2233 Homework Set 7
Math 33 Homework Set 7 1. Find the general solution to the following differential equations. If initial conditions are specified, also determine the solution satisfying those initial conditions. a y 4
More information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
More informationSeries Solutions of ODEs. Special Functions
C05.tex 6/4/0 3: 5 Page 65 Chap. 5 Series Solutions of ODEs. Special Functions We continue our studies of ODEs with Legendre s, Bessel s, and the hypergeometric equations. These ODEs have variable coefficients
More informationThe Generating Functions for Pochhammer
The Generating Functions for Pochhammer Symbol { }, n N Aleksandar Petoević University of Novi Sad Teacher Training Faculty, Department of Mathematics Podgorička 4, 25000 Sombor SERBIA and MONTENEGRO Email
More informationPUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes
PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES Notes. x n+ = ax n has the general solution x n = x a n. 2. x n+ = x n + b has the general solution x n = x + (n )b. 3. x n+ = ax n + b (with a ) can be
More informationMultiple Zeta Values of Even Arguments
Michael E. Hoffman U. S. Naval Academy Seminar arithmetische Geometrie und Zahlentheorie Universität Hamburg 13 June 2012 1 2 3 4 5 6 Values The multiple zeta values (MZVs) are defined by ζ(i 1,..., i
More informationarxiv: v1 [math-ph] 15 May 2012
All solutions of the n = 5 Lane Emden equation Patryk Mach arxiv:05.3480v [math-ph] 5 May 0 M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Abstract All
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationHorst Alzer A MEAN VALUE INEQUALITY FOR THE DIGAMMA FUNCTION
Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 75, 2 (207), 9 25 Horst Alzer A MEAN VALUE INEQUALITY FOR THE DIGAMMA FUNCTION Abstract. A recently published result states that for all ψ is greater than or
More informationA class of trees and its Wiener index.
A class of trees and its Wiener index. Stephan G. Wagner Department of Mathematics Graz University of Technology Steyrergasse 3, A-81 Graz, Austria wagner@finanz.math.tu-graz.ac.at Abstract In this paper,
More informationSome bounds for the logarithmic function
Some bounds for the logarithmic function Flemming Topsøe University of Copenhagen topsoe@math.ku.dk Abstract Bounds for the logarithmic function are studied. In particular, we establish bounds with rational
More informationIntroduction to Spherical Harmonics
Introduction to Spherical Harmonics Lawrence Liu 3 June 4 Possibly useful information. Legendre polynomials. Rodrigues formula:. Generating function: d n P n x = x n n! dx n n. wx, t = xt t = P n xt n,
More informationBoundary Value Problems in Cylindrical Coordinates
Boundary Value Problems in Cylindrical Coordinates 29 Outline Differential Operators in Various Coordinate Systems Laplace Equation in Cylindrical Coordinates Systems Bessel Functions Wave Equation the
More informationTriangle diagrams in the Standard Model
Triangle diagrams in the Standard Model A. I. Davydychev and M. N. Dubinin Institute for Nuclear Physics, Moscow State University, 119899 Moscow, USSR Abstract Method of massive loop Feynman diagrams evaluation
More informationSome Fun with Divergent Series
Some Fun with Divergent Series 1. Preliminary Results We begin by examining the (divergent) infinite series S 1 = 1 + 2 + 3 + 4 + 5 + 6 + = k=1 k S 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + = k=1 k 2 (i)
More informationSPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS
SPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS Second Edition LARRY C. ANDREWS OXFORD UNIVERSITY PRESS OXFORD TOKYO MELBOURNE SPIE OPTICAL ENGINEERING PRESS A Publication of SPIE The International Society
More informationd 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.
4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal
More informationCertain Generating Functions Involving Generalized Mittag-Leffler Function
International Journal of Mathematical Analysis Vol. 12, 2018, no. 6, 269-276 HIKARI Ltd, www.m-hiari.com https://doi.org/10.12988/ijma.2018.8431 Certain Generating Functions Involving Generalized Mittag-Leffler
More informationSome identities related to Riemann zeta-function
Xin Journal of Inequalities and Applications 206 206:2 DOI 0.86/s660-06-0980-9 R E S E A R C H Open Access Some identities related to Riemann zeta-function Lin Xin * * Correspondence: estellexin@stumail.nwu.edu.cn
More informationHigher Monotonicity Properties of q-gamma and q-psi Functions
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 8, Number 2, pp. 247 259 (213) http://campus.mst.edu/adsa Higher Monotonicity Properties of q-gamma and q-psi Functions Mourad E. H.
More information