Closed Form Solutions for Linear Differential and Difference Equations

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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.