On second order q-difference equations for high-order Sobolev-type q-Hermite orthogonal polynomials

TL;DR

This paper investigates higher-order q-Hermite I-Sobolev type polynomials, providing their hypergeometric representation, structure relations, recurrence relations, and establishing two different q-difference equations they satisfy.

Contribution

It introduces new q-difference equations for high-order Sobolev-type q-Hermite polynomials and details their structural and recurrence properties.

Findings

Hypergeometric representation of the polynomials

Derivation of structure relations and recurrence relations

Establishment of two q-difference equations

Abstract

The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.