Series expansions of multi-dimensional Mellin convolution integrals
José Luis López, Pedro J. Pagola · Integral Transforms and Special Functions · 2014
In a recent paper [Lopez JL. Asymptotic expansions of Mellin convolution integrals. SIAM Rev. 2008;50(2):275–293], we have presented a new, very general and simple method for deriving asymptotic expansions of for small x. In this paper we generalize that method to multi-dimensional integrals of the form . In this multi-dimensional case we require for f an extra condition not required in the one-dimensional case, namely, f must satisfy a certain homogeneity property in its variables. Watson's lemma for multiple integrals is obtained as a corollary. An asymptotic expansion of the Lauricella's function FA for large values of its variables is given as an illustration. The possible application of the method to study the asymptotic behaviour near the origin of double Mellin–Barnes type convolution integrals [Yakubovich SB. A constructive method for constructing integral convolutions. (Russian) Dokl Akad Nauk BSSR. 1990;34(7):588–591] is also indicated.