Closed Form Solutions for Linear Differential and Difference Equations

线性微分方程组和差分方程组出现在数学及其应用中的各种问题中。这项研究的主要目标是开发一个完整的算法，找到这类方程的所有闭合形式解。闭合形式解不是解的近似，而是由一个精确的公式给出的，它以众所周知的特殊函数的形式写成.这项研究将以新的方式结合多种技术，从局部数据(每个奇点的渐近行为)中获得关于闭合形式解的关键信息，这涉及到有趣的数论和组合问题.这种新方法最重要的特点是，它将导致一个可证明完整的算法来寻找所有闭合形式的解。

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.