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

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Abstract
An explicit form and its analytic continuation around the origin of the Borel sum of the Barnes generalized hypergeometric series {}_qF_{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.