Software for the Algorithmic Work with Orthogonal Polynomials and Special Functions

In the last decade major steps towards an algorithmic treatment of orthogonal polynomials and special functions (OP & SF) have been made, notably Zeilberger's brilliant extension of Gosper's algorithm on algorithmic definite hypergeometric summation. By implementations of these and other algorithms symbolic computation has the potential to change the daily work of everybody who uses orthogonal polynomials or special functions in research or applications. It can be expected that symbolic computation will also play an important role in on-line versions of major revisions of existing formula books in the area of OP & SF. It this couple of talks I present software in Maple of those algorithmic techniques, in particular of Gosper's, Zeilberger's, and Petkovsek's algorithms and their q-analogoues. Some implementational details are discussed. The main emphasis, however, is given to on-line demonstrations of these algorithms using our Maple implementations (jointly with Harald Boeing) covering many examples from the field of OP & SF.Comment: 31 pages, 3 figures, Plenary Talk at the IWOP 98, Madrid, June 29-30, 199

Efficient Rational Creative Telescoping

We present a new algorithm to compute minimal telescopers for rational functions in two discrete variables. As with recent reduction-based approach, our algorithm has the nice feature that the computation of a telescoper is independent of its certificate. Moreover, our algorithm uses a sparse representation of the certificate, which allows it to be easily manipulated and analyzed without knowing the precise expanded form. This representation hides potential expression swell until the final (and optional) expansion, which can be accomplished in time polynomial in the size of the expanded certificate. A complexity analysis, along with a Maple implementation, suggests that our algorithm has better theoretical and practical performance than the reduction-based approach in the rational case