Liouvillian solutions of irreducible linear difference equations

Abstract

In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. Compared to the prior algorithm by Hendriks and Singer, our main contribution consists of two theorems that significantly reduce the number of combinations that the algorithm will check.

Solving recurrence relations using local invariants

“…In our previous paper [7] the table consisted of a single equation, namely τ n + b, but this equation contained a parameter b ∈ C(x), so this one equation represents an infinite set of equations, parametrized by b ∈ C(x). Now before we can compute, if it exists, a transformation between the input equation L and an equation of the form τ n + b, we first have to compute the unknown b ∈ C(x).…”

“…Now before we can compute, if it exists, a transformation between the input equation L and an equation of the form τ n + b, we first have to compute the unknown b ∈ C(x). Most of [7] is devoted to finding a finite list of candidates for b.…”

Liouvillian solutions of irreducible second order linear difference equations

“…[7,5,6] resp. [11,12,3,14]. The algorithms for hypergeometric solutions use a combinatorial search, where each of the combinations involves computing polynomial solutions.…”

“…The algorithms for hypergeometric solutions use a combinatorial search, where each of the combinations involves computing polynomial solutions. The algorithms for Liouvillian solutions are also combinatorial in nature, either because they call an algorithm for hypergeometric solutions [11,12,3], or perform a reduced (but still exponential) combinatorial search [14].…”

“…Prior algorithms for Liouvillian solutions will also solve this equation, using a search that depends exponentially on the number of finite singularities (see [14] for more details). This paper introduces a new method for finding Liouvillian solutions.…”