Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme

TL;DR

This paper establishes bilateral discrete and continuous orthogonality relations within the $q^{-1}$-symmetric Askey scheme, deriving a $q$-beta integral and its Ramanujan-type limit with two proofs, advancing the understanding of these special functions.

Contribution

It introduces new bilateral orthogonality relations in the $q^{-1}$-Askey scheme and derives a novel $q$-beta integral with a Ramanujan-type limit, including two proofs.

Findings

Derived bilateral orthogonality relations for $q^{-1}$-polynomials.

Established a $q$-beta integral from orthogonality relations.

Connected the $q$-beta integral to Ramanujan-type beta integrals in the limit.

Abstract

In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a $q$-beta integral which comes from the continuous orthogonality relation for the $q^{-1}$-Askey--Wilson polynomials. In the $q\to 1^{-}$ limit, this $q$-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.