The Appell’s function $F\textunderscore \{2\}$ for large values of its variables

Abstract

Abstract.The second Appell's hypergeometric function F 2 (a, b, b , c, c ; x, y) has a Mellin convolution integral representation in the region (x + y) < 1 and a > 0. We apply a recently introduced asymptotic method for Mellin convolution integrals to derive three asymptotic expansions of F 2 (a, b, b , c, c ; x, y) in decreasing powers of x and y with x/y bounded. For certain values of the real parameters a, b, b , c and c , two of these expansions involve logarithmic terms in the asymptotic variables x and y. Some coefficients of these expansions are given in terms of the Gauss hypergeometric function 3 F 2 and its derivatives.