On the evaluation of the Appell $F_2$ double hypergeometric function

Abstract

The transformation theory of the Appell F2(a, b1, b2; c1, c2; x, y) double hypergeometric function is used to obtain a set of series representations of F2 which provide an efficient way to evaluate F2 for real values of its arguments x and y and generic complex values of its parameters a, b1, b2, c1 and c2 (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of F2 and other double hypergeometric series that appear in the intermediate steps of the calculations are written as infinite sums of one variable hypergeometric series, such as the Gauss 2F1 or the 3F2, various linear transformations of the latter being then applied to derive known and new formulas. Using the three well-known Euler transformations of F2 on these results allows us to obtain a total of 44 series which form the basis of the Mathematica package AppellF2, dedicated to the evaluation of F2. A brief description of the package and of the numerical analysis that we have performed to test it are also presented.

Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions

“…For the case of analytic continuations of H 5 , more work is needed to cover the whole space of its arguments, since white zones still appear. The analytic continuations of other order two complete double hypergeometric functions can be found by using their relation to the Appell F 2 function, as discussed in [27].…”

“…While all the four Appell functions are implemented in Maple, only the Appell F 1 function is implemented in the kernel of Mathematica (there are also external packages for F 1 [30,31]). The case of Appell F 2 has been recently considered by the present authors in [27] for real values of its arguments, in non logarithmic situations. We plan to built a numerical package for all the order two complete double variable hypergeometric functions and Lauricella functions in three variables, valid for complex values of their arguments, in the near future.…”

“…This approach was later used to the study of the three variables extension of F 1 , namely the Lauricella F (3) D , and even of its n-variable extension F (n) D by Exton [17]. In a similar way, Exton obtained the analytic continuations of the Appell F 4 series in [26] while the authors of the present paper, in collaboration with Marichev, have considered Olsson's approach in detail on the case of the Appell F 2 [27]. Olsson also studied F 2 in [28], although in the latter work the F 2 function is expressed as a Laplace transform of the product of two 1 F 1 functions (and thus, the linear transformations of 1 F 1 are used to find the analytic continuations of F 2 ).…”

“…We now demonstrate the working principle of each of these options. Although the Olsson command works for multivariable hypergeometric series, for the sake of illustration we take here as input the Appell F 2 double hypergeometric series, which we have recently considered in detail in [27]. The default value of each of the options above is False.…”

Linear transformations of Srivastava's $H_C$ triple hypergeometric function

“…One of the simplest possibilities is to write the studied hypergeometric function of n variables as an infinite sum of hypergeometric series with n − 1 variables and to apply the known transformation theory of the latter. This approach has been followed in a considerable number of publications (we refer the reader to [1] for a non-exhaustive list) and it has been revisited recently, in the linear case, by one of the authors of the present paper and collaborators, for the realization of a Mathematica package dedicated to the numerical evaluation of the Appell F 2 double hypergeometric function [2]. In addition, a first automatization of this linear transformation procedure has been presented in [3].…”