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

Gergő Nemes

TL;DR

This work extends hyperasymptotics to inverse factorial series solutions of high-order linear difference equations with polynomial coefficients and a rank-one irregular singularity at infinity. It uses Mellin-Borel transforms and a universal family of hyperterminants to build level-by-level hyperasymptotic expansions that uniquely determine solutions and facilitate numerical computation of Stokes multipliers (connection coefficients). The paper provides explicit error bounds for the inverse factorial expansions and demonstrates the framework on a Gauss hypergeometric function with a large parameter and on a third-order difference equation, including concrete coefficient computations and numerical results. The approach yields a robust, computable scheme to refine asymptotics in difference equations, with potential extensions to more general coefficient structures and higher-rank singularities, thereby enhancing both theoretical understanding and practical calculation of connection data.

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.

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

TL;DR

This work extends hyperasymptotics to inverse factorial series solutions of high-order linear difference equations with polynomial coefficients and a rank-one irregular singularity at infinity. It uses Mellin-Borel transforms and a universal family of hyperterminants to build level-by-level hyperasymptotic expansions that uniquely determine solutions and facilitate numerical computation of Stokes multipliers (connection coefficients). The paper provides explicit error bounds for the inverse factorial expansions and demonstrates the framework on a Gauss hypergeometric function with a large parameter and on a third-order difference equation, including concrete coefficient computations and numerical results. The approach yields a robust, computable scheme to refine asymptotics in difference equations, with potential extensions to more general coefficient structures and higher-rank singularities, thereby enhancing both theoretical understanding and practical calculation of connection data.

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.

Paper Structure

This paper contains 14 sections, 8 theorems, 175 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Borel transforms and their associated functions
  3. Hyperasymptotic expansions
  4. Superasymptotics
  5. Level-one
  6. Level-two
  7. General level
  8. Computation of the connection coefficients
  9. Error bounds for the inverse factorial series
  10. Applications
  11. Gauss hypergeometric function with a large third parameter
  12. A third-order difference equation
  13. Discussion
  14. Computation of the hyperterminants

Key Result

Proposition 2.1

Let $\eta \in \mathbb R$ be an admissible direction, and let $\mu_j$$(1\le j\le n)$ be given by mueq. For each $1\le j\le n$, the function $y_j(t,\eta)$, defined by is analytic in $\mathcal{P}_\eta$, satisfies the relation where the $K_{\ell,j }$ are constants, and admits endless analytic continuation along any path that does not contain any of the points $\lambda_1, \lambda_2, \ldots, \lambda_n

Figures (1)

  • Figure 1: Magnitudes of the level-two hyperasymptotic terms of $w_1(z,0)$ at $z = 30 + \mathrm{i}$, normalised by scaling with $2^{-z}\Gamma (z+\tfrac{1}{2})$.

Theorems & Definitions (17)

  • Remark
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop1']}
  • proof : Proof of Proposition \ref{['prop2']}
  • proof : Proof of Proposition \ref{['prop3']}
  • Theorem 3.1
  • Remark
  • Theorem 3.2
  • ...and 7 more