The Extended Zeilberger's Algorithm with Parameters

TL;DR

This paper extends Zeilberger's algorithm to handle multiple hypergeometric sums with parameters, enabling the derivation of parameter relations and recurrence relations for orthogonal polynomials, including their $q$-analogues.

Contribution

It introduces a generalized algorithm for hypergeometric sums with multiple terms and parameters, including $x$-free coefficient relations and applications to orthogonal polynomials.

Findings

Derived linear relations among multiple hypergeometric sums.

Enabled determination of three-term recurrence relations for orthogonal polynomials.

Extended the approach to $q$-analogues for Askey-Wilson and $q$-Racah polynomials.

Abstract

For a hypergeometric series $\sum_k f(k,a, b, ...,c)$ with parameters $a, b, >...,c$, Paule has found a variation of Zeilberger's algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general problem concerning several similar hypergeometric terms $f_1(k, a, b,..., c)$, $f_2(k, a,b, ..., c)$, $...$, $f_m(k, a, b, ..., c)$. We present an algorithm to derive a linear relation among the sums $\sum_k f_i(k,a,b,...,c)$ $(1\leq i \leq m)$. Furthermore, when the summand $f_i$ contains the parameter $x$, we can require that the coefficients be $x$-free. Such relations with $x$-free coefficients can be used to determine whether a polynomial sequence satisfies the three term recurrence and structure relations for orthogonal polynomials. The $q$-analogue of this approach is called the extended $q$-Zeilberger's algorithm, which can be employed to derive recurrence relations on the Askey-Wilson polynomials and the $q$-Racah polynomials.