A class of q-orthogonal polynomial sequences that extends the q-Askey scheme

TL;DR

This paper introduces explicit formulas for recurrence coefficients of a broad class of q-orthogonal polynomials extending the q-Askey scheme, unifying known families and discovering potential new sequences.

Contribution

It provides a unified parametric framework for q-orthogonal polynomials, encompassing all q-Askey scheme families and revealing new polynomial sequences.

Findings

Formulas express recurrence coefficients via four parameters.

Special parameter choices recover all q-Askey scheme polynomials.

Potentially new polynomial families are identified.

Abstract

We obtain new explicit formulas for the recurrence coefficients of the q-orthogonal polynomial sequences in a class that extends the q-Askey scheme. Our formulas express the recurrence coefficients in terms of four parameters that determine the zeroes and poles. By direct substitution of particular values for the four parameters we obtain all the polynomial sequences in the q-Askey scheme, including the Askey-Wilson and the Racah polynomials. We also obtain some families of sequences that may be new.