The goal in this paper is to find closed form solutions for linear recurrence equations, by transforming an input equation L to an equation Ls with known solutions. The main problem is how to find a solved equation Ls to which L can be reduced. We solve this problem by computing local data at singularities, data that remains invariant under the transformations used.

In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. The first algorithm for this was given by Hendriks in 1998, and Hendriks and Singer in 1999. Several improvements have been published, including a paper by Cha and van Hoeij that reduces the combinatorial problem. But the number of combinations still depended exponentially on the number of singularities. For irreducible second order equations, we give a short and very efficient algorithm; the number of combinations is 1.

The software to be presented is an implementation of the algorithms in [1], [2], and [3]. (This software is available at [4].) The main algorithm in [3] is currently implemented with additional base equations beyond what appear in [3] and will be part of the software demonstration in July.Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations). Before calling any of these algorithms we first check currently available algorithms for a solution in the form of a first order right hand factor or some other general term solution.Algorithm 'Find 2 F 1 ' will find a gt-transformation to a recurrence relation satisfied by a hypergeometric series u(n) = 2 F 1 a+n b c z , if such a transformation exists. As an example, sequence A005572 = [1, 4, 17, 76, 354, 1704, 8421, . . . ] from the OEIS ([5]) represents the "Number of walks on cubic lattice starting and finishing on the xy plane and never going below it." A005572 has offset 0 (i.e. the first entry in the list is A005572(0)) and satisfies: 12(n + 1)A005572(n) − (20 + 8n)A005572(n + 1) + (n + 4)A005572(n + 2) = 0.The output from our program is:The algorithm 'Find Liouvillian' will find a gt-transformation to a recurrence relation of the form u(n + 2) + b(n)u(n) = 0 for some b(n) ∈ C(n), if such a transformation exists. 'Find Liouvillian' is not unique in terms of its purpose but, for second order recurrence relations, it is faster than prior algorithms. As an example, for the recurrence relation satisfied by A099364 from the OEIS, our program produces the solution:

We will present an implementation of several algorithms for solving second order linear recurrence relations. The algorithms are described in two papers accepted at ISSAC 2010. Our implementation can find Liouvillian solutions, as well as solutions written in terms of values of special functions such as the 2F1 hypergeometric function, Bessel, Whittaker, Legendre, Laguerre, etc. We have done an automated search in Sloane's online encyclopedia of integer sequences, to find sequences that satisfy a second order recurrence. Our implementation solves a large majority of such recurrence relations. The papers and implementation are available at http://www.math.fsu.edu/~glevy/implementation.