Algorithms for q-Hypergeometric Summation in Computer Algebra

This paper describes three algorithms for q -hypergeometric summation: ? a multibasic analogue of Gosper?s algorithm, ? the q - Zeilberger algorithm, and ? an algorithm for finding q - hypergeometric solutions of linear recurrences together with their Maple implementations, which is relevant both to people being interested in symbolic computation and in q -series. For all these algorithms, the theoretical background is already known and has been described, so we give only short descriptions, and concentrate ourselves on introducing our corresponding Maple implementations by examples. Each section is closed with a description of the input/output specifications of the corresponding Maple command. We present applications to q -analogues of classical orthogonal polynomials. In particular, the connection coefficients between families of q -Askey?Wilson polynomials are computed. Mathematica implementations have been developed for most of these algorithms, whereas to our knowledge only Zeilberger?s algorithm has been implemented in Maple so far (Koornwinder, 1993 or Zeilberger, cf. Pe kov0sek et al., 1996). We made an effort to implement the algorithms as efficient as possible which in the q -Petkov?ek case led us to an approach with equivalence classes. Hence, our implementation is considerably faster than other ones. Furthermore the q -Gosper algorithm has been generalized to also find formal power series solutions.

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