Hypergeometric structures in Feynman integrals

Abstract

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Appendices

Appendix A: The multiple series representation

In the following we summarize the different multiple series representations that can be found in the existing literature. We note that starting with their partial linear differential representation our methods described above can also provide the presented sum representations. The expansion coefficients are given as rational functions of Pochhammer symbols, which requires a corresponding re-parameterization of the coefficients of the foregoing differential and difference equations.

One of the simplest functions of these classes is Gauß’ hypergeometric function (2). Its generalizations, the generalized hypergeometric functions, are given by (3). The bi–variate Appell series [8, 9] have the representations (4) and

$$ \begin{array}{@{}rcl@{}} F_{2}(a;b,b';c,c';x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m+n} (b)_{m} (b')_{n}}{m! n! (c)_{m} (c')_{n}} x^{m} y^{n} \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} F_{3}(a,a';b,b';c;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m} (a')_{n} (b)_{m} (b')_{n}}{m! n! (c)_{m+n}}x^{m} y^{n} \end{array} $$

(2)

$$ \begin{array}{@{}rcl@{}} F_{4}(a;b;c,c';x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m+n} (b)_{m+n}}{m! n! (c)_{m} (c')_{n}}x^{m} y^{n}. \end{array} $$

(3)

This set is extended to the Horn-type bi–variate series [12, 13]

$$ \begin{array}{@{}rcl@{}} G_{1}(a;b,b';x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} (a)_{m+n} (b)_{n-m} (b')_{m-n} \frac{x^{m} y^{n}}{m! n!} \end{array} $$

(4)

$$ \begin{array}{@{}rcl@{}} G_{2}(a,a';b,b';x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} (a)_{m} (a')_{n} (b)_{n-m} (b')_{m-n} \frac{x^{m} y^{n}}{m! n!} \end{array} $$

(5)

$$ \begin{array}{@{}rcl@{}} G_{3}(a,a';x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} (a)_{2n-m} (a')_{2m-n} \frac{x^{m} y^{n}}{m! n!} \end{array} $$

(6)

$$ \begin{array}{@{}rcl@{}} H_{1}(a;b;c;d;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m-n} (b)_{m+n} (c)_{n}}{(d)_{m}} \frac{x^{m} y^{n}}{m! n!} \end{array} $$

(7)

$$ \begin{array}{@{}rcl@{}} {H_{2}(a;b;c;d;e;x,y)} &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} { \frac{(a)_{m-n} (b)_{m} (c)_{n} (d)_{n} }{(e)_{m}} \frac{x^{m} y^{n}}{m! n!} } \end{array} $$

(8)

$$ \begin{array}{@{}rcl@{}} H_{3}(a;b;c;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{2m+n} (b)_{n}}{(c)_{m+n}}\frac{x^{m} y^{n}}{m! n!} \end{array} $$

(9)

$$ \begin{array}{@{}rcl@{}} H_{4}(a;b;c;d;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{2m+n} (b)_{n}}{(c)_{m} (d)_{n}}\frac{x^{m} y^{n}}{m! n!} \end{array} $$

(10)

$$ \begin{array}{@{}rcl@{}} H_{5}(a;b;c;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{2m+n} (b)_{n-m}}{(c)_{n}}\frac{x^{m} y^{n}}{m! n!} \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} H_{6}(a;b;c;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} (a)_{2m-n} (b)_{n-m} (c)_{n} \frac{x^{m} y^{n}}{m! n!} \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} H_{7}(a;b;c;d;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{2m-n} (b)_{n} (c)_{n}}{(d)_{m}} \frac{x^{m} y^{n}}{m! n!}. \end{array} $$

(13)

As in the case of the generalized hypergeometric functions there are confluent forms of the bi–variate functions.

There are also the further bi–variate series [18, 19]

$$ \begin{array}{@{}rcl@{}} S_{1}(a,a',b;c,d;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m+n} (a')_{m+n} (b)_{m}}{(c)_{m+n} (d)_{m}} \frac{x^{m} y^{n}}{m! n!}. \end{array} $$

(14)

$$ \begin{array}{@{}rcl@{}} S_{2}(a,a',b, b';c;x,y) &=& \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^{\infty} \frac{(a)_{m-n} (a')_{m-n} (b)_{n} (b')_{m}}{(c)_{m-n}} \frac{x^{m} y^{n}}{m! n!}. \end{array} $$

(15)

The generalization of the bi–variate hypergeometric functions is the Kampé de Fériet function [11]

$$ \begin{array}{@{}rcl@{}} F_{C;D;D'}^{A;B:B'}\biggl[\genfrac..{0pt}{}{a,b,b'}{c,d,d'};x,y\biggr] = \sum\limits_{m,n = 0}^{\infty} \frac{ (a_{1})_{m+n} ... (a_{A})_{m+n} (b_{1})_{m} ... (b_{B})_{m} (b'_{1})_{n} ... (b'_{B'})_{n}} {(c_{1})_{m+n} ... (c_{C})_{m+n} (d_{1})_{m} ... (d_{D})_{m} (d'_{1})_{n} ... (d'_{D'})_{n}} \frac{x^{m} y^{n}}{m! n!}. \end{array} $$

(16)

Triple hypergeometric functions of the Lauricella–Saran type [21,22,23] are

$$ \begin{array}{@{}rcl@{}} \hat{F}_{4} &=& F_{E}(a_{1},a_{1},a_{1},b_{1},b_{2};c_{1},c_{2},c_{3};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m+n+p} (b_{1})_{m} (b_{2})_{n+p}}{(c_{1})_{m} (c_{1})_{n} (c_{1})_{p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{14} &=& F_{F}(a_{1},a_{1},a_{1},b_{1},b_{2},b_{1};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m+n+p} (b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{8} &=& F_{G}(a_{1},a_{1},a_{1},b_{1},b_{2},b_{3};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m+n+p} (b_{1})_{m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(19)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{3} &=& F_{K}(a_{1},a_{2},a_{2},b_{1},b_{2},b_{1};c_{1},c_{2},c_{3};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m} (a_{2})_{n+p}(b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m} (c_{2})_{n} (c_{3})_{p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(20)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{11} &=& F_{M}(a_{1},a_{2},a_{1},b_{1},b_{2},b_{1};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m} (a_{2})_{n+p}(b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(21)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{6} &=& F_{N}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{1};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m} (a_{2})_{n} (a_{3})_{p} (b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!} \\ \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{12} &=& F_{P}(a_{1},a_{2},a_{1},b_{1},b_{1},b_{2};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m+p} (a_{2})_{n} (b_{1})_{m+n} (b_{2})_{p}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(23)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{10} &=& F_{P}(a_{1},a_{2},a_{1},b_{1},b_{2},b_{1};c_{1},c_{2},c_{2};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m+p} (a_{2})_{n} (b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m} (c_{2})_{n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(24)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{7} &=& F_{P}(a_{1},a_{2},a_{2},b_{1},b_{2},b_{3};c_{1},c_{1},c_{1};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m} (a_{2})_{n+p} (b_{1})_{m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{m+n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}\\ \end{array} $$

(25)

$$ \begin{array}{@{}rcl@{}} \hat{F}_{13} &=& F_{T}(a_{1},a_{2},a_{2},b_{1},b_{2},b_{1};c_{1},c_{1},c_{1};x,y,z) = \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a_{1})_{m} (a_{2})_{n+p} (b_{1})_{m+p} (b_{2})_{n}}{(c_{1})_{m+n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}.\\ \end{array} $$

(26)

There are three more triple hypergeometric functions, the Srivastava functions [20],

$$ \begin{array}{@{}rcl@{}} \hat{H}_{A}(a,b,b';c,c';x,y,z) &=& \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a)_{m+p} (b)_{m+n} (b')_{n+p}}{(c)_{m} { (c')}_{n+p} } \frac{x^{m} y^{n} z^{p}}{m! n! p!} \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} \hat{H}_{B}(a,b,b';c_{1},c_{2},c_{3};x,y,z) &=& \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a)_{m+p} (b)_{m+n} (b')_{n+p}}{(c_{1})_{m} (c_{2})_{n} (c_{3})_{p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!} \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} \hat{H}_{C}(a,b,b';c;x,y,z) &=& \sum\limits_{m=0,n=0,p=0}^{\infty} \frac{(a)_{m+p} (b)_{m+n} (b')_{n+p}}{(c)_{m+n+p}} \frac{x^{m} y^{n} z^{p}}{m! n! p!}. \end{array} $$

(29)

These functions are given in the literature in different forms. The following identities hold comparing to Ref. [20]:

$$ \begin{array}{@{}rcl@{}} \hat F_{4}(a_{1},b_{1},b_{2};c_{1};x,y,z) &\to& f_{20a}(a_{1},b_{2},b_{1};c_{1};z,y,x) \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} \hat F_{14}(a_{1},b_{1},b_{2};c_{1};x,y,z) &\to& f_{21a}(a_{1},b_{1},b_{2};c_{1},c_{2};x,z,y) \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} \hat F_{8}(a_{1},b_{1},b_{2},b_{3};c_{1},c_{2};x,y,z) &\to& f_{18a}(a_{1},b_{3},b_{2},b_{1};c_{2},c_{1};z,y,x) \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} \hat F_{3}(a_{1},a_{2},b_{1},b_{2};c_{1},c_{2},c_{3};x,y,z)&\to& f_{10a}(a_{2},b_{1},b_{2},a_{1};c_{2},c_{3},c_{1};y,z,x) \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} \hat F_{11}(a_{1},a_{2},b_{1},b_{2};c_{1},c_{2};x,y,z) &\to& f_{11a}(a_{2},b_{1},b_{2},a_{1};c_{2},c_{1},y,z,x) \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} \hat F_{6}(a_{1},a_{2},a_{3},b_{1},b_{2};c_{1},c_{2};x,y,z)&\to& f_{6a}(b_{1},a_{1},a_{3},a_{2},b_{1};c_{1},c_{2};x,z,y) \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} \hat F_{12}(a_{1},a_{2},b_{1},b_{2};c_{1},c_{2};x,y,z) &\to& f_{12a}(b_{1},a_{1},a_{2},b_{2};c_{2},c_{1};y,x,z) \end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} \hat F_{10}(a_{1},a_{2},b_{1},b_{2};c_{1},c_{2};x,y,z) &\to& f_{9a}(b_{1},a_{1},a_{2},b_{2};c_{1},c_{2};x,z,y) \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} \hat F_{7}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3};c_{1};x,y,z)&\to& f_{7a}(a_{2},b_{2},b_{3},b_{1},a_{1};c_{1};y,z,x) \end{array} $$

(38)

$$ \begin{array}{@{}rcl@{}} \hat F_{13}(a_{1},a_{2},b_{1},b_{2};c_{1};x,y,z) &\to& f_{13a}(a_{2},b_{1},b_{2},a_{1};c_{1};y,z,x) \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} \hat H_{A}(a,b,b_{1};c,c_{1};x,y,z) &\to& f_{15a}(b_{1},a,b;c_{1}{,c};y,z,x) \end{array} $$

(40)

$$ \begin{array}{@{}rcl@{}} \hat H_{B}(a,b,b_{1};c_{1},c_{2},c_{3};x,y,z) &\to& f_{14a}(b,b_{1},a;c_{1},c_{2},c_{3};x,y,z) \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} \hat H_{C}(a,b,b_{1};c;x,y,z) &\to& f_{16a}(b,b_{1},a;c;x,y,z). \end{array} $$

(42)

A comprehensive list of triple hypergeometric series f_1b to f_62b has been given in Ref. [20, 169,170,171,172,173,174,175,176,177,178,179].

The quadruple hypergeometric functions by Exton [14] are¹

$$ \begin{array}{@{}rcl@{}} K_{1} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m+n} (b_{2})_{p}}{(c_{1})_{l+p} (c_{2})_{m} (c_{3})_{n}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} K_{2} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m+n} (b_{2})_{p}}{(c_{1})_{l} (c_{2})_{m} (c_{3})_{n} (c_{4})_{p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} K_{3} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n+p}}{(c_{1})_{l+p} (c_{2})_{m+n}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(45)

$$ \begin{array}{@{}rcl@{}} K_{4} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n+p}}{(c_{1})_{l+p} (c_{2})_{m} (c_{3})_{n}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(46)

$$ \begin{array}{@{}rcl@{}} K_{5} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n+p}}{(c_{1})_{l} (c_{2})_{m} (c_{3})_{n} (c_{4})_{p} } \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} K_{6} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{l} (c_{2})_{m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} K_{7} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{l+n} (c_{2})_{m+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(49)

$$ \begin{array}{@{}rcl@{}} K_{8} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{l+n} (c_{2})_{m} (c_{3})_{p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(50)

$$ \begin{array}{@{}rcl@{}} {K_{9}} &=& { \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{l} (c_{2})_{m} (c_{3})_{n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!}} \end{array} $$

(51)

$$ \begin{array}{@{}rcl@{}} K_{10} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l+m} (b_{2})_{n} (b_{3})_{p}}{(c_{1})_{l} (c_{2})_{m} (c_{3})_{n} (c_{4})_{p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(52)

$$ \begin{array}{@{}rcl@{}} K_{11} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p}}{(c_{1})_{l+m+n} (c_{2})_{p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(53)

$$ \begin{array}{@{}rcl@{}} K_{12} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p}}{(c_{1})_{l+m} (c_{2})_{n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(54)

$$ \begin{array}{@{}rcl@{}} K_{13} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n+p} (b_{1})_{l} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p}}{(c_{1})_{l+m} (c_{2})_{n} (c_{3})_{p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(55)

$$ \begin{array}{@{}rcl@{}} K_{14} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n} (b_{1})_{p} (b_{2})_{l+p} (b_{3})_{m} (b_{4})_{n}}{({c})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(56)

$$ \begin{array}{@{}rcl@{}} K_{15} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m+n} (b_{1})_{p} (b_{2})_{l} (b_{3})_{m} (b_{4})_{n} (b_{5})_{p}}{(c_{1})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(57)

$$ \begin{array}{@{}rcl@{}} K_{16} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{({a_{1}})_{l+m} ({a_{2}})_{l+n} ({a_{3}})_{m+p} ({a_{4}})_{n+p}}{({c})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(58)

$$ \begin{array}{@{}rcl@{}} K_{17} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m} (b_{1})_{l+n} (b_{2})_{m+n} (b_{3})_{p} (b_{4})_{p}}{({c})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(59)

$$ \begin{array}{@{}rcl@{}} K_{18} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m} (b_{1})_{l+p} (b_{2})_{m+n} (b_{3})_{n} (b_{4})_{p}}{(c_{1})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(60)

$$ \begin{array}{@{}rcl@{}} K_{19} &=& \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m} (b_{1})_{l+n} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p} (b_{5})_{p}}{(c_{1})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!} \end{array} $$

(61)

$$ \begin{array}{@{}rcl@{}} K_{20} &=& { \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a_{1})_{l+m} (a_{2})_{n+p} (b_{1})_{l} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p}}{(c_{1})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!}} \end{array} $$

(62)

$$ \begin{array}{@{}rcl@{}} K_{21} &=& { \sum\limits_{l,m,n,p = 0}^{\infty} \frac{(a)_{l+m} (b_{1})_{l} (b_{2})_{m} (b_{3})_{n} (b_{4})_{p} (b_{5})_{n} (b_{6})_{p}}{(c_{1})_{l+m+n+p}} \frac{x^{l} y^{m} z^{n} t^{p}}{l! m! n! p!}.} \end{array} $$

(63)

Furthermore, there are the multivariate Lauricella functions [21]

$$ \begin{array}{@{}rcl@{}} F_{A}^{(n)}(a,b_{1},...,b_{n};c_{1},...,c_{n};x_{1},...,x_{n}) &=& \sum\limits_{m_{1},...,m_{n}=0}^{\infty} \frac{(a)_{m_{1}+...+m_{n}} (b_{1})_{m_{1}} ... (b_{n})_{m_{n}}}{(c_{1})_{m_{1}} ... (c_{n})_{m_{n}}} \frac{x_{1}^{m_{1}} ... x_{n}^{m_{n}}}{m_{1}! ... m_{n}!}\\ \end{array} $$

(64)

$$ \begin{array}{@{}rcl@{}} F_{B}^{(n)}(a_{1} ... a_{n},b_{1},...,b_{n};c_{1},...,c_{n};x_{1},...,x_{n}) &=& \sum\limits_{m_{1},...,m_{n}=0}^{\infty} \frac{(a_{1})_{m_{1}} ... (a_{n})_{m_{n}} (b_{1})_{m_{1}} ... (b_{n})_{m_{n}}}{(c_{1})_{m_{1}} ... (c_{n})_{m_{n}}} \frac{x_{1}^{m_{1}} ... x_{n}^{m_{n}}}{m_{1}! ... m_{n}!}\\ \end{array} $$

(65)

$$ \begin{array}{@{}rcl@{}} F_{C}^{(n)}(a,b;c_{1},...,c_{n};x_{1},...,x_{n}) &=& \sum\limits_{m_{1},...,m_{n}=0}^{\infty} \frac{ (a)_{m_{1}+ ... m_{n}} (b)_{m_{1}+ ... m_{n}}}{(c_{1})_{m_{1}} ... (c_{n})_{m_{n}}} \frac{x_{1}^{m_{1}} ... x_{n}^{m_{n}}}{m_{1}! ... m_{n}!} \end{array} $$

(66)

$$ \begin{array}{@{}rcl@{}} F_{D}^{(n)}(a,b_{1} ... b_{n};c;x_{1},...,x_{n}) &=& \sum\limits_{m_{1},...,m_{n}=0}^{\infty} \frac{ (a)_{m_{1}+ ... m_{n}} (b_{1})_{m_{1}} ... (b_{n})_{m_{n}}}{(c)_{m_{1} + ... m_{n}}} \frac{x_{1}^{m_{1}} ... x_{n}^{m_{n}}}{m_{1}! ... m_{n}!}.\\ \end{array} $$

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The file cases.m gives an even more extensive computer–readable list of these functions.

Appendix B: Mapping conditions to the Pochhammer case

The parameters of the general representations of the (partial) differential equations given in Section 2, leading to product solutions, obey the well–know Pochhammer solutions, if they apply a number of relations. Here we present some typical examples for these relations. The case of two variables:

$$ \begin{array}{@{}rcl@{}} F_{1} &:& \{a\}\to \alpha \beta ,\{b\}\to \alpha +\beta +1,\{c,c_{1}\}\to -\gamma ,\{d,g,d_{1},g_{1}\}\to -1,\{e,h,e_{1},h_{1}\}\to 1,\\ && \{f\}\to \beta ,\{j,j_{1}\}\to 0,\{a_{1}\}\to \alpha \beta_{1},\{b_{1}\}\to \alpha +\beta_{1}+1,\{f_{1}\}\to \beta_{1}\} \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} H_{1} &:& \{a\}\to \alpha \beta ,\{b\}\to \alpha +\beta +1,\{c\}\to -\delta ,\{d,j,g_{1}\}\to -1,\{e,d_{1},e_{1},h_{1}\}\to 1,\\ && \{f\}\to \alpha -\beta -1,\{g,h,j_{1}\}\to 0,\{a_{1}\}\to \beta \gamma ,\{b_{1}\}\to \beta +\gamma +1,\{c_{1}\}\to 1-\alpha ,\\ &&\{f_{1}\}\to \gamma \end{array} $$

(69)

$$ \begin{array}{@{}rcl@{}} S_{1} &:& \{ a \} \to \alpha \alpha_{1} \beta; \{ b \} \to (1 + \alpha_{1}) (1 + \beta) + \alpha (1 + \alpha_{1} + \beta); \{ c \} \to -(\gamma \delta);\\ && \{ d \} \to -1 - \gamma - \delta; \{ e \} \to 3 + \alpha + \alpha_{1} + \beta; \{ f \} \to (1 + \alpha + \alpha_{1}) \beta; \{ g \} \to -\delta;\\ &&\{ h \} \to 3 + \alpha + \alpha_{1} + 2 \beta; \{ j \} \to \beta; \{ l, q, d_{1}, g_{1} \} \to -1; \{ p, s, e_{1}, j_{1} \} \to 1;\\ &&\{ r, h_{1} \} \to 2; \{ a_{1} \} \to \alpha \alpha_{1}; \{ b_{1}, f_{1} \} \to 1 + \alpha + \alpha_{1}; \{ c_{1} \} \to -\gamma. \end{array} $$

(70)

The case of three variables, [20]:

$$ \begin{array}{@{}rcl@{}} f_{1b} &:& \{ A \} \to b_{1} b_{2}; \{ B_{0} \} \to -c; \{ B_{1} \} \to 1 + b_{1} + b_{2}; \{ C_{1} \} \to -b_{2}; \{ D_{1} \} \to -b_{1};\\ && \{ E_{0}, H_{1}, L_{1}, H'_{2}, {L}^{\prime\prime}_{2} \} \to -1; \{ E_{1}, S_{1}, F'_{0}, F'_{1}, {G}^{\prime\prime}_{0}, {G}^{\prime\prime}_{1} \} \to 1; \{ F_{1}, G_{1}, H_{0}, L_{0}, B'_{1}, D'_{1},\\ && E'_{1}, G'_{1}, H'_{1}, L'_{1}, S'_{0}, S'_{1}, {B}^{\prime\prime}_{1}, {C}^{\prime\prime}_{1}, {E}^{\prime\prime}_{1}, {F}^{\prime\prime}_{1}, {H}^{\prime\prime}_{1}, {L}^{\prime\prime}_{1}, {S}^{\prime\prime}_{1}, {S}^{\prime\prime}_{2} \} \to 0; \{ A' \} \to a_{1} a_{2};\\ && \{ C'_{0} \} \to 1 - b_{1}; \{ C'_{1} \} \to 1 + a_{1} + a_{2}; \{ {A}^{\prime\prime} \} \to a_{3} a_{4}; \{ {D}^{\prime\prime}_{0} \} \to 1 - b_{2};\\ && \{ {D}^{\prime\prime}_{1} \} \to 1 + a_{3} + a_{4} \end{array} $$

(71)

$$ \begin{array}{@{}rcl@{}} f_{21a} &:& \{ A, A' \} \to a_{1} a_{2}; \{ B_{0} \} \to -c_{1}; \{ B_{1}, C_{1}, B'_{1}, C'_{1} \} \to 1 + a_{1} + a_{2}; \{ D_{1}, D'_{1} \} \to a_{2};\\ && \{ E_{0}, F'_{0}, S'_{0}, {G}^{\prime\prime}_{0}, {S}^{\prime\prime}_{2} \} \to -1; \{ E_{1}, F_{1}, L_{1}, S_{1}, E'_{1}, F'_{1}, L'_{1}, S'_{1}, {G}^{\prime\prime}_{1}, {L}^{\prime\prime}_{1}, {S}^{\prime\prime}_{1} \} \to 1;\\ && \{ G_{1}, H_{0}, L_{0}, G'_{1}, H'_{2}, {E}^{\prime\prime}_{1}, {F}^{\prime\prime}_{1}, {H}^{\prime\prime}_{1}, {L}^{\prime\prime}_{2} \} \to 0; \{ H_{1}, H'_{1} \} \to 2; \{ C'_{0}, {D}^{\prime\prime}_{0} \} \to -c_{2};\\ && \{ {A}^{\prime\prime} \} \to a_{1} a_{3}; \{ {B}^{\prime\prime}_{1}, {C}^{\prime\prime}_{1} \} \to a_{3}; \{ {D}^{\prime\prime}_{1} \} \to 1 + a_{1} + a_{3} \end{array} $$

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$$ \begin{array}{@{}rcl@{}} f_{27b} &:& \{ A \} \to a_{1} a_{2}; \{ B_{0}, C'_{0} \} \to 1 - b; \{ B_{1} \} \to 1 + a_{1} + a_{2}; \{ C_{1} \} \to a_{2}; \{ D_{1}, F_{1}, G_{1},\\ && L_{1}, S_{1}, D'_{1}, E'_{1}, G'_{1}, L'_{1}, S'_{1}, {L}^{\prime\prime}_{2}, {S}^{\prime\prime}_{2} \} \to 0; \{ E_{0}, E_{1}, H_{0}, H_{1}, F'_{0}, F'_{1}, H'_{1}, H'_{2}, {E}^{\prime\prime}_{1}, {F}^{\prime\prime}_{1} \} \to 1;\\ && \{ L_{0}, S'_{0} \} \to -2; \{ A' \} \to a_{1} a_{3}; \{ B'_{1} \} \to a_{3}; \{ C'_{1} \} \to 1 + a_{1} + a_{3}; \{ {A}^{\prime\prime} \} \to b (1 + b);\\ && \{ {B}^{\prime\prime}_{1}, {C}^{\prime\prime}_{1} \} \to -2 b; \{ {D}^{\prime\prime}_{0} \} \to -c; \{ {D}^{\prime\prime}_{1} \} \to 2 (3 + 2 b);\\ && \{ {G}^{\prime\prime}_{0} \} \to -1; \{ {G}^{\prime\prime}_{1} \} \to 4; \{ {H}^{\prime\prime}_{1} \} \to 2; \{ {L}^{\prime\prime}_{1}, {S}^{\prime\prime}_{1} \} \to -4. \end{array} $$

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The case of four variables, [14, 15]:

$$ \begin{array}{@{}rcl@{}} K_{1} &:& \{ A, A', {A}^{\prime\prime} \} \to a b_{1}; \{ B_{0}, {E}^{\prime\prime\prime}_{0} \} \to -c_{1}; \{ B_{1}, C_{1}, D_{1}, B'_{1}, C'_{1}, D'_{1}, {B}^{\prime\prime}_{1}, {C}^{\prime\prime}_{1}, \\&& {D}^{\prime\prime}_{1} \} \to 1 + a + b_{1}; \{ E_{1}, E'_{1}, {E}^{\prime\prime}_{1} \} \to b_{1}; \{ F_{0}, P_{0}, G'_{0}, {H}^{\prime\prime}_{0}, {L}^{\prime\prime\prime}_{0}, {P}^{\prime\prime\prime}_{2} \} \to -1; \\&& \{ F_{1}, G_{1}, H_{1}, P_{1}, R_{1}, S_{1}, F'_{1}, G'_{1}, H'_{1}, P'_{1}, R'_{1}, S'_{1}, {F}^{\prime\prime}_{1}, {G}^{\prime\prime}_{1}, {H}^{\prime\prime}_{1}, {P}^{\prime\prime}_{1}, {R}^{\prime\prime}_{1}, {S}^{\prime\prime}_{1}, {L}^{\prime\prime\prime}_{1}, {P}^{\prime\prime\prime}_{1}, \\&& {R}^{\prime\prime\prime}_{1}, {S}^{\prime\prime\prime}_{1} \} \to 1; \{ L_{1}, M_{0}, N_{0}, L'_{1}, M'_{2}, Q'_{0}, R'_{0}, {L}^{\prime\prime}_{1}, {N}^{\prime\prime}_{2}, {Q}^{\prime\prime}_{2}, {S}^{\prime\prime}_{0}, {F}^{\prime\prime\prime}_{1}, {G}^{\prime\prime\prime}_{1}, {H}^{\prime\prime\prime}_{1}, {M}^{\prime\prime\prime}_{1}, \\&& {N}^{\prime\prime\prime}_{1}, {Q}^{\prime\prime\prime}_{1}, {R}^{\prime\prime\prime}_{2}, {S}^{\prime\prime\prime}_{2} \} \to 0; \{ M_{1}, N_{1}, Q_{1}, M'_{1}, N'_{1}, Q'_{1}, {M}^{\prime\prime}_{1}, {N}^{\prime\prime}_{1}, {Q}^{\prime\prime}_{1} \} \to 2; \{ C'_{0} \} \to -c_{2}; \\&& \{ {D}^{\prime\prime}_{0} \} \to -c_{3}; \{ {A}^{\prime\prime\prime} \} \to a b_{2}; \{ {B}^{\prime\prime\prime}_{1}, {C}^{\prime\prime\prime}_{1}, {D}^{\prime\prime\prime}_{1} \} \to b_{2}; \{ {E}^{\prime\prime\prime}_{1} \} \to 1 + a + b_{2} \end{array} $$

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$$ \begin{array}{@{}rcl@{}} { K_{21} }&:& \{ A \} \to a b_ 1; \{ B_ 0, C'_ 0, {D}^{\prime\prime}_ 0, {E}^{\prime\prime\prime}_ 0 \} \to -c_ 1; \{ B_ 1 \} \to 1 + a + b_ 1; \{ C_ 1 \} \to b_ 1; \\&& \{ D_ 1, E_ 1, G_ 1, H_ 1, L_ 1, N_ 1, P_ 1, Q_ 1, R_ 1, S_ 1, D'_ 1, E'_ 1, F'_ 1, H'_ 1, L'_ 1, N'_ 1, P'_ 1, Q'_ 1, R'_ 1, S'_ 1, {B}^{\prime\prime}_ 1, {C}^{\prime\prime}_ 1, \\&& {E}^{\prime\prime}_ 1, {F}^{\prime\prime}_ 1, {G}^{\prime\prime}_ 1, {L}^{\prime\prime}_ 1, {M}^{\prime\prime}_ 1, {N}^{\prime\prime}_ 1, {P}^{\prime\prime}_ 1, {Q}^{\prime\prime}_ 1, {R}^{\prime\prime}_ 1, {S}^{\prime\prime}_ 1, {B}^{\prime\prime\prime}_ 1, {C}^{\prime\prime\prime}_ 1, {D}^{\prime\prime\prime}_ 1, {F}^{\prime\prime\prime}_ 1, {G}^{\prime\prime\prime}_ 1, {H}^{\prime\prime\prime}_ 1, {M}^{\prime\prime\prime}_ 1, {N}^{\prime\prime\prime}_ 1, {P}^{\prime\prime\prime}_ 1, {Q}^{\prime\prime\prime}_ 1, \\&& {R}^{\prime\prime\prime}_ 1, {S}^{\prime\prime\prime}_ 1 \} \to 0; \{ F_ 0, M_ 0, N_ 0, P_ 0, G'_ 0, M'_ 2, Q'_ 0, R'_ 0, {H}^{\prime\prime}_ 0, {N}^{\prime\prime}_ 2, {Q}^{\prime\prime}_ 2, {S}^{\prime\prime}_ 0, {L}^{\prime\prime\prime}_ 0, {P}^{\prime\prime\prime}_ 2, {R}^{\prime\prime\prime}_ 2, {S}^{\prime\prime\prime}_ 2 \} \to -1; \\&& \{ F_ 1, M_ 1, G'_ 1, M'_ 1, {H}^{\prime\prime}_ 1, {L}^{\prime\prime\prime}_ 1 \} \to 1; \{ A' \} \to a b_ 2; \{ B'_ 1 \} \to b_ 2; \{ C'_ 1 \} \to 1 + a + b_ 2; \\&& \{ {A}^{\prime\prime} \} \to b_ 3 b_ 5; \{ {D}^{\prime\prime}_ 1 \} \to 1 + b_ 3 + b_ 5; \{ {A}^{\prime\prime\prime} \} \to b_ 4 b_{6}; \{ {E}^{\prime\prime\prime}_ 1 \} \to 1 + b_ 4 + b_{6} \}. \end{array} $$

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The complete list of relations is given in computer readable form in the attachment Mconditions.m to this paper.

Appendix C: A brief descriptions of the commands of HypSeries

In the following we describe the commands available in the Mathematica package HypSeries. To execute this package requires also the packages Sigma, EvaluateMultiSums [125,126,127,128], and HarmonicSums as well as other packages, see Appendix F. The user has to provide n (partial) differential equations in the n–variable case. The commands

$$ {\texttt {solveDE1}},~~ {\texttt {solveDE2}},~~ {\texttt {solveDE3}},~~ {\texttt {solveDE4}} $$

check whether the corresponding set of one to four variables has product solutions by consulting internal lists of cases. If applicable, the corresponding product solution for the expansion coefficients f(m) to f(m,n,p,q) are provided. In the bi–variate case subclasses are dealt with individually. The command is e.g.

$${\texttt {solveDE4[\{eq1==0,eq2==0,eq3==0,eq4==0\},\{x,y,z,t\},\{m,n,p,q\}]}}.$$

More general solutions are possible by using the command DEProductSolution. One has to provide the required n differential equations in the list

$$ {\texttt{sys = \{eq1==0,...,eqn==0\}}}.$$

Then

$$ {\texttt{DEProductSolution[sys,\{x,y,...\},\{m,n,...\}]}} $$

returns the respective expansion coefficient f(m,n,p,q). Here the tools described in Section 3.1 are utilized.

If one has, on the reverse, a Pochhammer ratio A = f(m,n,p,q) the command

$$ {\texttt{findDE[A,\{x,y,...\},\{m,n,...\}]}} $$

returns the system of differential equations obeyed by

$$ f(x,y,...) = \sum\limits_{m,n,...\ge =0}^{\infty} f(m,n,...) x^{m} y^{n} \cdots $$

Given a differential equation equ in n variables the command findRE

$$ {\texttt{findRE[eq==0,\{x,y,\ldots\},\{m,n,\ldots\}]}} $$

returns a corresponding recurrence for f(m,n,...). The last two commands implement the techniques presented in the beginning of Section 3.

To prepare for the expansion in the dimensional parameter ε, which frequently occurs in the parameters of the differential and difference equations, one usually needs to check for the convergence domain of the corresponding solution, to be able to perform the respective limit $N \rightarrow \infty$ in the sums involved. Generally it is assumed that {x,y,z,t}∈] − 1, 1[. However, often stronger conditions are needed in the multivariate cases. Internal Tables, cf. [20], allow to check for this in the bi– and tri–variate cases using the commands findCond2 and findCond3. One first has to determine the corresponding function label fcn via classifier2, classifier3, as e.g.

$$ \texttt {classifier3[f[m,n,p], \{x,y,z\}, \{m,n,p\}}] $$

returning fcn. Then

$$ {\texttt{ findCond3[fcn,\{x,y,z\}}}] $$

returns the convergence conditions, which are in some cases given in implicit form. The ε–expansion is performed using algorithms implemented in Sigma. The attached notebooks ExHypSeries.nb and ExSolvePartialLDE.nb give a more detailed explanation on this.

In the cases wherein the ε–expansion of the considered higher transcendental functions can be performed to a certain power O(ε^k) one may want to check, whether the solution satisfies the corresponding differential equations. This is provided by the command CheckDE[sol,eq], where sol denotes the solution up to the corresponding degree in ε and eq the differential equation:

$${\texttt {CheckDE[sol,eq]}}$$

returns then a result, which is of higher order in ε.

Appendix D: A brief descriptions of the commands of solvePartialLDE

The Mathematica package SolvePLDE.m implements the aforementioned algorithms for solving partial linear difference equations. It requires Sigma and HarmonicSums to be loaded. Additionally, the software Singular [180] must be installed, and made available through the interface given in [181]. The installation path of Singular can be set using the command appropriate for the user’s system, e.g.

< <Singular.m

SingularCommand = " (path to) /Singular-4.1.3-x86_64-Linux/bin /Singular".

The functions available are

• spread[p,q,{n, k,...}(,{eps,...})]: this function calculates the spread of the polynomials p and q, in the variables n,k,…. The symbols in the optional list are treated as an extension to the field over which the polynomials are defined. If the polynomials p and q contain symbolic parameters other than n,k,…, such as for instance the dimensional regulator ε, they must be declared in the second list.

• dispersion[p,q,{n, k,...}(,{eps,...})]: this function calculates the dispersion (it is the maximum of the spread) of the polynomials p and q in the variables n,k,…. The second optional list has the same function as in the function spread.

• SolvePLDE[eq==rhs,f[n,k,...],(options)]. This command solves the linear partial difference equation. It has the following available options:

  • UseObject →list of Harmonic sums and/or Pochhammer symbols Allows to define a list of harmonic sums and Pochhammer symbols to be searched in the numerator of the solution.

  • PLDEdegBound →d Allows to choose the total degree d of the Ansatz for the numerator of the solution. Defaults to 0.

  • InsertDenFactor →factors In the case the periodic denominator bound was not complete, the user may force the search to include factors in the denominator.

  • PLDESymbols →list Any symbols appearing other than the shift variables must be declared in list.

  • InitialValues →list A list of initial values in the form $\{\{var_{1}\rightarrow val_{1}, var_{2}\rightarrow val_{2},\ldots , initial value\},\ldots \}$

  • SymbolDegree →d When initial conditions are provided, a linear combination of the homogeneous solutions is built, having as coefficients rational functions in the symbols. This option sets the maximum total degree of the numerator and denominator of those rational functions.

• $\texttt {SolveExpand[eq==rhs,f[n,k,...],PLDEExpandIn}\rightarrow \{\varepsilon ,\ell _{min},\ell _{max}\},$ $\texttt {InitialValues}\rightarrow \{\ldots \},(options)]$: this command solves the PLDE in a series expansion in a parameter. The options are the same as for SolvePLDE.

• expandHypergPref[eq==rhs, f[n,k,...],fac]. This command derives a new equation whose solution has the hypergeometric factor fac removed, as described in Section 6.3.1.

Appendix E: A constant

We calculate the constant C given in (115). The two contributions to this infinite sum do both diverge, while

$$ \begin{array}{@{}rcl@{}} C \approx 2.759413418790153909406713643175. \end{array} $$

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Unfortunately (115) cannot simply be written in terms of a hypergeometric function of main argument 1, since this diverges, which is easily seen by applying Gauß’ formula. However, one can define it as the following limit

$$ \begin{array}{@{}rcl@{}} C &=& \lim_{\varepsilon \rightarrow 0} \left\{1 - _{2}F_{1}\left( \frac{1}{2}-\frac{i \sqrt{3}}{2},\frac{1}{2}+\frac{i \sqrt{3}}{2};1;1-\varepsilon\right)-\frac{\cosh \left[\frac{\sqrt{3} \pi }{2}\right] \ln (\varepsilon)}{\pi }\right\}. \end{array} $$

(77)

Before expanding in ε one should map the main argument as, cf. e.g. [7],

$$ \begin{array}{@{}rcl@{}} 1 - \varepsilon \rightarrow \frac{1}{\varepsilon} - 1. \end{array} $$

(78)

Furthermore, the relation for the digamma function

$$ \begin{array}{@{}rcl@{}} \psi(1-z) = \psi(z) + \pi \cot(\pi z) \end{array} $$

(79)

shall be used. One finally obtains (116). In deriving (116) one obtains first an expression both containing the real and the imaginary part of $\psi \left (\frac {1}{2}+\frac {i\sqrt {3}}{2}\right )$. Those obey the following integral representations

$$ \begin{array}{@{}rcl@{}} \Re\left[\psi\left( \frac{1}{2}+\frac{i\sqrt{3}}{2}\right)\right] &=& {{\int}_{0}^{1}} dt \left[\frac{ t^{-1/2} \cos\left[\frac{\sqrt{3}}{2} \ln(t) \right]}{t-1} - \frac{1}{\ln(t)}\right] \end{array} $$

(80)

$$ \begin{array}{@{}rcl@{}} \Im\left[\psi\left( \frac{1}{2}+\frac{i\sqrt{3}}{2}\right)\right] &=& {{\int}_{0}^{1}} \frac{dt}{\sqrt{t}(t-1)} \sin\left[\frac{\sqrt{3}}{2} \ln(t) \right] = \frac{\pi}{2} \tanh\left[\frac{\sqrt{3}}{2} \pi \right]. \end{array} $$

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While the imaginary part of $\psi \left (\frac {1}{2}+\frac {i\sqrt {3}}{2}\right )$ evaluates to a basic function, a further simplification seems not to be possible.

The new entities emerging here are therefore

$$ \begin{array}{@{}rcl@{}} e^{\pi \sqrt{3}/2},~~~~~~~ \text{and}~~~~~~~ \psi\left( \frac{1}{2}+\frac{i\sqrt{3}}{2}\right). \end{array} $$

(82)

In the expansion of multi–variate series of the Pochhammer-type one expects quite new classes of special numbers to emerge, which will become a potential research topic in the future.

Appendix F: Software required

The ancillary files cover the Mathematica notebooks ExHypSeries.nb and ExSolvePartialLDE.nb . Their execution requires the following software packages, which can be downloaded from the software site of the RISC institute:

EvaluateMultiSums.m https://risc.jku.at/sw/evaluatemultisums/ Guess.m, LinearSystemSolver.m https://risc.jku.at/sw/guess/ HarmonicSums.m https://risc.jku.at/sw/harmonicsums/ Sigma.m https://risc.jku.at/sw/sigma/

In addition, the computer algebra system Singular Singular.m http://www.singular.uni-kl.de has to be installed in order to execute the Mathematica package SolvePartialLDE.m.

All data and its supplementary information files generated or analyzed during this study are included in this published article:

cases.m converg.m HypSeries.m Mconditions.m SolvePartialLDE.m ExHypSeries.nb ExSolvePartialLDE.nb .

It is recommended to download all precomputed tables for HarmonicSums. The Mathematica notebooks ExHypSeries.nb and ExsolvePartialLDE.nb contain the information from where the correct version of the used packages can be downloaded.

While the notebook ExsolvePartialLDE.nb needs a few minutes computation time only, ExHypSeries.nb needs 1.32 days.