A~new theory of multiple hypogeometric series and its prospects for computer algebra programming

A. W. Niukkanen

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 3, pp. 717-745

Résumé

We first give a brief overview of basic concepts and calculational techniques of the operator factorization method which provides a simple and universal foundation for a new theory of hypergeometric series in any number of variables. Notation, factorization formulas and all auxiliary identities used in the paper are given in section 2, 3 and 4, respectively. The most essential constituents are: illustrative examples showing how the method works in practice (section 5); new transformation theory of multiple hypergeometric series (section 6); the prospects of the method including applications to computer algebra systems and discussion of the position occupied by the new method amongst other approaches to the theory of hypergeometric series (section 7). The conclusion of practical importance is that the factorization method is the only method allowing us to develop active knowledge-based computer programs rather than the passive data bases typical for traditional approaches. In other words the new method holds out a hope for programs capable to synthesize the formulas at the moment they are needed by the user.