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

Abstract

The transformation theory of the Appell $F_2(a,b_1,b_2;c_1,c_2;x,y)$ double hypergeometric function is used to obtain a set of series representations of $F_2$ which provide an efficient way to evaluate $F_2$ for real values of its arguments $x$ and $y$ and generic complex values of its parameters $a,b_1, b_2, c_1$ and $c_2$ (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of $F_2$ 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 $_2F_1$ or the $_3F_2$, various linear transformations of the latter being then applied to derive known and new formulas. Using the three well-known Euler transformations of $F_2$ 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 $F_2$. A brief description of the package and of the numerical analysis that we have performed to test it are also presented.