Stavros Garoufalidis
Abstract: 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 Andre.
Key words: $G$-functions, holonomic functions, holonomic sequences, $D$-finite sequences, Zeilberger, hypergeometric terms, quasi-unipotent monodromy, asymptotic expansions, Gevrey series, Apery sequence.
Notes: 8 pages, no figures.