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The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of $G$-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms.
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O. Costin and S. Garoufalidis, Resurgence of the Kontsevich-Zagier power series
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S. Garoufalidis, An ansatz for the asymptotics of hypergeometric multisums
2008
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