An ansatz for the asymptotics of hypergeometric multisums

TL;DR

This paper introduces a new method for analyzing the asymptotic behavior of sequences defined by hypergeometric multisums, bypassing traditional recurrence-based approaches by directly identifying singularities through algebraic and geometric structures.

Contribution

The authors propose a novel shortcut for asymptotic analysis by constructing a finite set of potential singularities directly from hypergeometric terms, supported by conjectures and proofs in special cases.

Findings

Finite set of singularities can be constructed from hypergeometric terms.

The finite set contains all singularities of the generating series in many cases.

Examples demonstrate the effectiveness of the proposed method.

Abstract

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive $K$-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.