Explicit matrix inverses for lower triangular matrices with entries involving continuous q-ultraspherical polynomials

TL;DR

This paper derives explicit inverses for a family of lower triangular matrices involving continuous q-ultraspherical polynomials, extending previous results and exploring applications and limits related to q-Hermite polynomials.

Contribution

It provides explicit inverse matrices involving continuous q-ultraspherical functions, using a different proof approach with q-Racah polynomials, extending known results.

Findings

Explicit inverse matrices involving q-ultraspherical polynomials

Connections to q-Racah polynomials for proofs

Limit case leading to continuous q-Hermite polynomials

Abstract

For a one-parameter family of lower triangular matrices with entries involving continuous $q$-ultraspherical polynomials we give an explicit lower triangular inverse matrix, with entries involving again continuous $q$-ultraspherical functions. The matrices are $q$-analogues of results given by Cagliero and Koornwinder recently. The proofs are not $q$-analogues of the Cagliero-Koornwinder case, but are of a different nature involving $q$-Racah polynomials. Some applications of these new formulas are given. Also the limit $\beta \to 0$ is studied and gives rise to continuous $q$-Hermite polynomials for $0 < q < 1$ and $q > 1$.