The Borel Sum of Divergent Barnes Hypergeometric Series and its Application to a Partial Differential Equation

Abstract

An explicit form and its analytic continuation around the origin of the Borel sum of the Barnes generalized hypergeometric series ${}_q{F}_{p-1}$ of divergent type ($q>p$) is obtained. As an application we give an integral representation of the Borel sum of the formal solution to the Cauchy problem of a certain partial differential equation of non-Kowalevski type.