G-functions and multisum versus holonomic sequences

Details G-functions and multisum versus holonomic sequences G-functions and multisum versus holonomic sequences

2007-08-31

arxiv.org/abs/0708.4354v3

Mathematics

Combinatorics

8 pages, no figures

Scientific paper

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. The proofs utilize the notion of a $G$-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by Andr\'e.

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