Algorithmic determination of q-power series for q-holonomic functions

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 c̀onvert/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 for q-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, Paule, and Petkovšek (1998)], [Böing and Koepf (1999)] and [Abramov, Petkovšek, and Ryabenko (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 classical q-orthogonal polynomials of the q-Hahn class [Koekoek and Swarttouw (1998)]. Then we develop an algorithm which converts a q-holonomic recurrence equation of a q-hypergeometric series with nontrivial expansion point into the corresponding q-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)].