On peculiar properties of generating functions of some orthogonal polynomials

TL;DR

This paper explores special generating function identities for q-Hermite and related orthogonal polynomials, revealing new interpretations and methods to generate broader families of orthogonal polynomials, with proofs for classical and free cases.

Contribution

It establishes new generating function identities involving q-Hermite, big q-Hermite, and Askey--Chihara polynomials, and relates them to classical orthogonal polynomials, proposing conjectures for Askey--Wilson polynomials.

Findings

Derived identities involving q-Hermite and big q-Hermite polynomials.

Connected orthogonal polynomials like Hermite, Chebyshev, and Laguerre through these identities.

Proved conjectures for q=1 and q=0 cases, supporting generalizations.

Abstract

We prove that for |x|,|t|<1, -1 <q \leq1 and n\geq0: \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{n+i}(x|q) = h_{n}(x|t,q) \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{i}(x|q), where h_{n}(x|q) and h_{n}(x|t,q) are respectively the so called q-Hermite and the big q-Hermite polynomials and (q)_{n} denotes the so called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam---Chihara (ASC) polynomials and ASC and the so called continuous dual q-Hahn (c2h) polynomials. Moreover we are able to relate in this way some other 'ordinary ' orthogonal polynomials such as e.g. Hermite, Chebyshev or Laguerre. These equalities give new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey--Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case) thus paving the way to generalization of AW polynomials at least in these two cases.