A $q$-generalization of the para-Racah polynomials

TL;DR

This paper introduces a new family of bispectral orthogonal polynomials derived from Askey-Wilson polynomials, generalizing para-Racah polynomials with explicit formulas, orthogonality, and connections to other polynomial families.

Contribution

It presents a novel q-generalization of para-Racah polynomials obtained via an unconventional truncation of Askey-Wilson polynomials, including explicit hypergeometric expressions and orthogonality relations.

Findings

Derived new bispectral orthogonal polynomials from Askey-Wilson polynomials.

Established explicit hypergeometric series expressions and orthogonality relations.

Connected the new polynomials with q-Racah and dual-Hahn polynomials.

Abstract

New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic bi-lattice. The three term recurrence relation and q-difference equation are obtained through limits of those of the Askey-Wilson polynomials. An explicit expression in terms of hypergeometric series and the orthogonality relation are provided. A $q$-generalization of the para-Krawtchouk polynomials is obtained as a special case. Connections with the $q$-Racah and dual-Hahn polynomials are also presented.