Computation of Mellin convolution integrals with a logarithmic kernel: application to the third Appell function

Abstract

In [López JL. Asymptotic expansions of Mellin convolution integrals. SIAM Rev. 2008;50(2):275–293] we introduced a method to compute Mellin convolution integrals of the form in terms of a double power series of x. In this paper we extend the method to convolution integrals with a logarithmic kernel of the form . As we did in [1], we only require for f(t) and h(t) to have a power asymptotic expansion at t=∞ and t=0, respectively. We apply this method to derive an asymptotic expansion of the third Appell function for one large variable. The accuracy of the approximation is illustrated with numerical experiments.