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The Borel sum of divergent Barnes hypergeometric series and its application to a partial differential equation. (English) Zbl 0969.33004

The hypergeometric series \(_qF_{q-1}(q>p)\) is divergent. An explicit formula for the Borel sum is obtained and the analytic continuation is developed around the origin. First the results are obtained for the case in which no pair of numerator parameters differ by an integer; the complications are discussed and illustrated for the relaxation of this condition. The Laplace integral, its inversion, and residue calculus are used in the development. Further, the connections with a Cauchy problem for a related partial differential equation are displayed.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
34E05 Asymptotic expansions of solutions to ordinary differential equations
40G10 Abel, Borel and power series methods
40A10 Convergence and divergence of integrals
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