Using symbolic computer algebraic systems to derive formulas involving orthagonal polynomials and other special functions

Author: George Gasper

Abstract:

It is shown how symbolic computer algebraic systems such as Mathematica, Macsyma, SMP, etc., can be used to derive transformation and expansion formulas for orthogonal polynomials that are expressible in terms of either hypergeometric or basic hypergeometric series. In particular, we demonstrate how Mathematica can be used to apply transformation formulas to the Racah and {\abit q}-Racah polynomials, to derive an indefinite bibasic summation formula, an expansion formula for Laguerre polynomials, Clausen's formula for the square of hypergeometric series, a {\abit q}-analogue of a Fields and Wimp expansion formula, and to prove the Askey-Gasper inequality which de Branges used in his proof of the Bieberbach conjecture. We also make some observations and conjectures related to Jensen's necessary and sufficient conditions for the Riemann Hypothesis to hold.

This article published in

Orthogonal Polynomials: Theory and Practice, ed. by P. Nevai, Kluwer Academic Publishers, Boston (1990), pp. 163 - 179.