q-special functions, basic hypergeometric series and operators

TL;DR

This paper explores $q$-hypergeometric series, difference equations, and operators, revealing their properties, eigenfunctions, and connections to orthogonal polynomials like little $q$-Jacobi polynomials within the $q$-Askey scheme.

Contribution

It provides a comprehensive analysis of $q$-hypergeometric operators, their factorizations, eigenfunctions, and extensions, including new insights into their orthogonality and transmutation properties.

Findings

Eigenfunctions are polynomials, specifically little $q$-Jacobi polynomials.

Operator factorization yields orthogonality and recurrence relations.

Connections between different $q$-hypergeometric operators are established via $q$-fractional derivatives.

Abstract

In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural Hilbert space --a weighted sequence space-- on which this operator is symmetric. Then the corresponding eigenfunctions are polynomials, which are the little $q$-Jacobi polynomials. These polynomials form a family in the $q$-Askey scheme, and so many important properties are well known. In particular, we show how the orthogonality relations and the three-term recurrence for the little $q$-Jacobi polynomials can be obtained using only the factorisation of the corresponding operator. As a next step we consider the $q$-hypergeometric operator in general, which leads to the little $q$-Jacobi functions. We sketch the derivation of the corresponding orthogonality using the connection between various eigenfunctions. The link between the $q$-hypergeometric operators with different parameters is studied in general using $q$-analogues of fractional derivatives, and this gives transmutation properties for this operator. In the final parts of these notes we consider partial extensions of this approach to orthogonal polynomials and special functions. The first extension is a brief introduction to the Askey-Wilson functions and the corresponding integral transform. The second extension is concerned with a matrix-valued extension of the $q$-hypergeometric difference equation and its solutions.