Closed Form Solutions for Linear Differential and Difference Equations

线性微分方程和差分方程出现在各种各样的数学问题及其应用中。本研究的主要目标是开发一个完整的算法来找到这些方程的所有闭形式解。封闭形式的解不是解的近似值，相反，它们是由一个精确的公式给出的，用众所周知的特殊函数来表示。这项研究将以新颖的方式结合各种各样的技术。封闭形式解的关键信息将以一种涉及有趣的数论和组合问题的新方法从局部数据（每个奇点的渐近行为）中获得。这种新方法最重要的特点是，它将导致一个可证明的完整算法，以找到所有封闭形式的解。

Linear differential equations and difference equations appear in a wide variety of problems in mathematics and its applications. The main goal of this research is to develop a complete algorithm that finds all closed form solutions of such equations. Closed form solutions are not approximations of solutions, instead, they are given by an exact formula, written in terms of well known special functions.This research will combine a wide variety of techniques in novel ways.Key information about closed form solutions will be obtained from local data (the asymptotic behavior at each singular point) in a new way that involves interesting number theoretical and combinatorial problems. The most important feature of this new approach is that it will lead to a provably complete algorithm to find all closed form solutions.