G-functions and multisum versus holonomic sequences

TL;DR

This paper explores the properties of sequences derived from hypergeometric sums, constructs related G-functions from combinatorics, and addresses a question about the representation of holonomic sequences, combining analytic and combinatorial methods.

Contribution

It proves that multidimensional sums of balanced hypergeometric terms have Gevrey-type asymptotic expansions, constructs new G-functions from combinatorics, and provides a counterexample to a question on holonomic sequences.

Findings

Sequences from hypergeometric sums have Gevrey-1 asymptotics with rational exponents.

Constructed G-functions originate from enumerative combinatorics.

Counterexample shows not all holonomic sequences are multisums of balanced hypergeometric terms.

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 Andr\'e.