Algorithmic Determination of q-Power Series for q-Holonomic Functions
Abstract
In [Koe1992] 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 and extend this algorithm in such a way that it identifies linear combinations of q-hypergeometric series. 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 [KS1998]. Then we develop a new 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 achieved. 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 more) 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 used to deduce special function identities in an easy way.
(jointly with Wolfram Koepf / Journal of Symbolic Computation 2011)
