Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations

This paper introduces a new concept of finite singularities that is useful for studying difference equations. For example, in theorems 2 and 3 I use this concept prove new results on the algebraic extensions one needs for solving operators of order 2 and 3. These two theorems are examples of what one can do with this concept of finite singularities. For the case order>3, and for more details and explicit examples on finite singularities, see the paper "Computing Hypergeometric Solutions of Linear Recurrence Equations" which can be found here.

The algorithm given in the paper has now been implemented. Here is a version for Maple6 (source code) and a comparison. Note that this code is now part of Maple 9 (hypergeomsols in LREtools).

For related ideas, additional explanations, and examples see:

Rational Solutions of Linear Difference Equations
Section 5 of Desingularization of linear difference operators with polynomial coefficients
The examples in Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients
The preprint "Computing Hypergeometric Solutions of Linear Recurrence Equations" which can be downloaded from here.

Download this paper.