On the evaluation of the Appell F2 double hypergeometric function

Abstract

The transformation theory of the Appell F2(a,b1,b2;c1,c2;x,y) double hypergeometric function is developed 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 F12 or the F23, various linear transformations of the latter being then applied to derive known and new formulas. Use of 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.wl, dedicated to the evaluation of F2. A brief description of the package and of the numerical analysis that we have performed to test it is also presented. Program summaryProgram Title: AppellF2.wlCPC Library link to program files:https://doi.org/10.17632/n9v6bwpsyd.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 11.3 and beyondNature of problem: Numerical evaluation of the double hypergeometric Appell F2 function for real values of its variables and generic complex values of its Pochhammer parametersSolution method: Mathematica implementation of a set of transformation formulas of the Appell F2 function