TAILIEUCHUNG - Báo cáo toán học: "A Generalization of Gosper’s Algorithm to Bibasic"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A Generalization of Gosper’s Algorithm to Bibasic . | A Generalization of Gosper s Algorithm to Bibasic Hypergeometric Summation Axel Riese Research Institute for Symbolic Computation Johannes Kepler University Linz A-4040 Linz Austria Submitted May 9 1996 Accepted June 24 1996. Abstract An algebraically motivated generalization of Gosper s algorithm to indefinite bibasic hypergeometric summation is presented. In particular it is shown how Paule s concept of greatest factorial factorization of polynomials can be extended to the bibasic case. It turns out that most of the bibasic hypergeometric summation identities from literature can be proved and even found this way. A Mathematica implementation of the algorithm is available from the author. AMS Subject Classification. Primary 33D65 68Q40 Secondary 33D20. 1 Introduction Recently Paule and Strehl 10 observed that the algorithm presented by Gosper 7 for indefinite hypergeometric summation extends quite naturally to the g-hypergeometric case by introducing a g-analogue of the canonical Gosper-Petkovsek GP representation for rational functions. Based on the new algebraic concept of greatest factorial factorization GFF Paule 8 developed an alternative but equivalent approach to hypergeometric telescoping. It was also shown by Paule cf. Paule and Riese 9 that the problem of g-hypergeometric telescoping can be treated along the same lines as the g 1 case by making use of a g-version of GFF. Built on these concepts a Mathematica implementation of g-analogues of Gosper s as well as of Zeilberger s 14 fast algorithm for definite g-hypergeometric summation has been carried out by the author cf. Paule and Riese 9 and Riese 12 . The original approach to definite g-hypergeometric summation is due to Wilf and Zeilberger 13 . The object of this paper is to describe how the algorithm gTelescope presented in 9 a g-analogue of Gosper s algorithm generalizes to the bibasic hypergeometric case. In Section 2 the underlying theoretical background based on a .

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