标题： 线性差分方程在一级非正则奇点处的超渐近分析：多项式系数 显示英文标题

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

摘要： 超渐近分析是一种将指数小项纳入渐近逼近的解析方法，从而扩展其适用范围，提高精度，并提供对底层奇点结构的更深入理解。 它还允许计算特定于问题的不变量，如斯托克斯乘子，这些值在其他方法中通常被假设或未知。 对于微分方程而言，与标准渐近展开不同，超渐近展开可以唯一确定解。 在本文中，我们将超渐近方法扩展到某些高阶线性差分方程的逆阶乘级数解，并证明所得展开式同样可以唯一确定解。 我们进一步说明如何使用超渐近技术数值计算这些展开式中出现的连接系数。 此外，我们给出了逆阶乘级数解的显式余项界。 我们的主要工具是梅林-博雷尔变换。 这些展开式通过与积分和微分方程上下文中熟悉的超终止函数密切相关的通用超终止函数来表达。 结果通过具有大第三个参数的高斯超几何函数和一个三阶差分方程进行了说明。 显示英文摘要

摘要： 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.