On the derivatives 2 P ν (z)/ ν 2 and Q ν (z)/ ν of the Legendre functions with respect to their degrees

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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).