Algorithmic determination of [formula omitted]-power series for [formula omitted]-holonomic functions

Abstract

In Koepf (1992) it was shown how for a given holonomic function a representation as a formal power series of hypergeometric type can be determined algorithmically. This algorithm–that we call FPS algorithm (Formal Power Series)–combines three steps to obtain the desired representation. The authors implemented this algorithm in the computer algebra system Maple as ‘convert/FormalPowerSeries’ which is always successful if the input function is a linear combination of hypergeometric power series.In this paper we give a q-analogue of the FPS algorithm forq-holonomic functions and extend this algorithm in such a way that it identifies and returns linear combinations of q-hypergeometric series. The algorithm is a combination of mainly three subalgorithms, which make use of existing algorithms from Abramov et al. (1998), Böing and Koepf (1999) and Abramov et al. (2000). We introduce two different polynomial bases for the representation of q-series and realize that they are sufficient to obtain all well-known q-hypergeometric representations of the classicalq-orthogonal polynomials of the q-Hahn class Koekoek and Swarttouw (1998). Then we develop an algorithm which converts aq-holonomic recurrence equation of a q-hypergeometric series with nontrivial expansion point into the correspondingq-holonomic recurrence equation for the coefficients. Furthermore, we show how the inverse problem can be handled. The latter algorithm is used to detect q-holonomic recurrences for some types of generalized q-hypergeometric functions. We implemented all presented algorithms (and many others) in Maple and make them available as Maple package qFPS which will be described briefly. Additionally, in some examples we show how qFPS can be applied to deduce special function identities in a simple way based on techniques used in Zeilberger (1990).