Solving linear recurrence equations

Abstract

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: