On the evaluation of the Appell F2 double hypergeometric function

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 2F1 or the 3F2, 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.本文围绕Appell F2(a, b1, b2; c1, c2; x, y)双超几何函数（Appell F2 double hypergeometric function）的变换理论展开研究，推导得到一组F2的级数表示形式，可高效计算当自变量x、y取实数值，参数a、b1、b2、c1及c2取任意复数值（即非对数情形）时的F2函数值。本研究采用经典研究框架，将F2及计算过程中间步骤中出现的其他双超几何级数的常规双级数表示，改写为单变量超几何级数（如高斯2F1（Gauss 2F1）或3F2）的无穷和形式，随后通过对上述单变量级数施加各类线性变换，推导出已知及全新的公式。基于上述结果，结合Appell F2的三类经典欧拉变换，最终共得到44个级数表示形式，以此构建了专用于F2函数求值的Mathematica软件包AppellF2.wl的核心内容。本文还简要介绍了该软件包，以及为验证其有效性所开展的数值分析工作。