The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations

TL;DR

This paper develops a new double-product representation for nonterminating basic hypergeometric series, explores properties of symmetric and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, and derives new generating functions and hypergeometric representations.

Contribution

It introduces the $q$-Chaundy theorem for hypergeometric series and applies it to derive novel properties, generating functions, and representations for symmetric and $q^{-1}$-symmetric orthogonal polynomials.

Findings

Derived double-product representations for hypergeometric series.

Obtained new generating functions for various $q$-orthogonal polynomials.

Established new hypergeometric and summation formulas for these polynomials.

Abstract

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the $q$-Chaundy theorem and several limiting $q\to 1^{-}$ cases are considered. Using the $q$-Chaundy theorem, we explore properties of the symmetric and $q^{-1}$-symmetric basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. These are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials. For instance, we show how many known (and unknown) generating functions can be easily derived for these polynomials. We also explore other methods to find generating functions for these polynomials. By applying the $q$-Chaundy theorem to the Ismail--Masson $q$-exponential generating function for continuous $q$ and $q^{-1}$-Hermite polynomials, we are able to derive alternative expansions of these generating functions, and from these, new terminating basic hypergeometric representations for the continuous $q$ and $q^{-1}$-Hermite polynomials. New quadratic transformations for the terminating basic hypergeometric series involved connect these representations. For the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials and as well their dual polynomials, which include the big and little $q$-Jacobi polynomials and the $q^{-1}$-Bessel polynomials, we discuss, and show how to exploit special orthogonality relations (integral and infinite series), connection formulas, and duality relations for these infinite families to derive new generating relations, and as well summation and integration formulas.