Tables of the Appell Hypergeometric Functions $F_2$

TL;DR

This paper develops new reduction formulas for the Appell hypergeometric function F2 by expressing it in terms of simpler hypergeometric functions, aiding in easier evaluation and application.

Contribution

The paper introduces new reduction formulas for F2, derived from existing $_2F_1$ and $_3F_2$ formulas, simplifying its computation.

Findings

New reduction formulas for F2 are tabulated.

F2 can be expressed in terms of $_2F_1$ and $_3F_2$ functions.

Simplifies evaluation of F2 for specific parameters.

Abstract

The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases of the generalized hypergeometric function $_qF_p$. The Appell hypergeometric functions $F_q$, $q=1,2,3,4$ are product of two hypergeometric functions $_2F_1$ that appear in many areas of mathematical physics. Here, we are interested in the Appell hypergeometric function $F_2$ which is known to have a double integral representation. As demonstrated by Opps, Saad, and Srivastava (J. Math. Anal. Appl. 302 (2005) 180-195), the double integral representation of $F_2$ can be reduced to a single integral that can be easily evaluated for certain values of the parameters in terms of $_2F_1$ and $_3F_2$. Using many of the reduction formulas of $_2F_1$ and $_3F_2$ and the representation of $F_2$ in terms of a single integral, we have begun to tabulate new reduction formulas for $F_2$.