Second structure relation for $q$-semiclassical polynomials of the Hahn Tableau

TL;DR

This paper derives a second structure relation for q-semiclassical orthogonal polynomials of the Hahn Tableau, extending the known first relation and providing new insights into their algebraic structure.

Contribution

The paper introduces a second structure relation for q-semiclassical polynomials, expanding the theoretical framework beyond the previously known first relation.

Findings

Derived a second structure relation for q-semiclassical polynomials

Established a finite-type relation between polynomial sequences and their q-differences

Enhanced understanding of the algebraic properties of q-semiclassical polynomials

Abstract

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.