The Appell ’ S Function F 2 for Large Values of Its Variables

The second Appell’s hypergeometric function F2(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 F2(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 3F2 and its derivatives.