Algorithmic determination of \(q\)-power series for \(q\)-holonomic functions (Q412219)

In [\textit{W. Koepf}, ``Power series in computer algebra'', J. Symb. Comput. 13, No. 6, 581--603 (1992; Zbl 0758.30026)], the Formal Power Series (FPS) algorithm was introduced. This algorithm, which is implemented in the computer algebra system Maple, can be used to find a representation as a formal power series for a given holonomic function. The algortihm is always successful if the input function is a linear combination of hypergeometric power series.NEWLINENEWLINEIn this paper the authors present a \(q\)-analogue of this FPS algorithm for \(q\)-holonomic functions. This algorithm is a combination of mainly three subalgorithms, which make use of existing algorithms earlier obtained by others.NEWLINENEWLINEThe \(q\)-FPS algorithm can be used to find all \(q\)-hypergeometric representations of the classical \(q\)-orthogonal polynomials belonging to the so-called \(q\)-Hahn class. Moreover, an algorithm is presented which converts a \(q\)-holonomic recurrence relation of a \(q\)-hypergeometric function into a corresponding \(q\)-holonomic recurrence relation for the coefficients. Furthermore, it is shown how the inverse problem can be handled. This latter algorithm can be used to detect \(q\)-holonomic recurrences for some types of generalized \(q\)-hypergeometric functions.NEWLINENEWLINEThe algorithms are implemented into a Maple package which can also be used to deduce \(q\)-special functions identities such as the \(q\)-Chu-Vandermonde summation formula.NEWLINENEWLINEThe paper is a condensed version of the PhD thesis by the first author.