Hyperasymptotics for linear difference equations with an irregular singularity of rank one: Polynomial coefficients

TL;DR

This paper extends hyperasymptotic methods to linear difference equations with irregular singularities, enabling precise solutions, computation of connection coefficients, and explicit bounds, with applications to special functions like the hypergeometric function.

Contribution

It introduces hyperasymptotic expansions for inverse factorial series solutions of higher-order difference equations, ensuring unique solution determination and providing numerical methods for connection coefficients.

Findings

Hyperasymptotics determine solutions uniquely for certain difference equations.

Explicit remainder bounds are derived for inverse factorial series solutions.

Connection coefficients can be computed numerically using hyperasymptotic techniques.

Abstract

Hyperasymptotics is an analytical method that incorporates exponentially small contributions into asymptotic approximations, thereby expanding their domain of validity, improving accuracy, and providing deeper insight into the underlying singularity structures. It also allows for the computation of problem-specific invariants, such as Stokes multipliers, whose values are often assumed or remain unknown in other approaches. For differential equations, unlike standard asymptotic expansions, hyperasymptotic expansions determine solutions uniquely. In this paper, we extend the hyperasymptotic method to inverse factorial series solutions of certain higher-order linear difference equations and demonstrate that the resulting expansions also determine the solutions uniquely. We further indicate how the connection coefficients appearing in these expansions can be computed numerically using hyperasymptotic techniques. In addition, we give explicit remainder bounds for the inverse factorial series solutions. Our main tool is the Mellin--Borel transform. The expansions are expressed via universal hyperterminant functions, closely related to the hyperterminants familiar from integral and differential equation contexts. The results are illustrated by the Gauss hypergeometric function with a large third parameter and a third-order difference equation.