The orthogonality of q-classical polynomials of the Hahn class: A geometrical approach

TL;DR

This paper presents a unified geometric approach to analyze the orthogonality of all positive definite q-polynomial solutions of the hypergeometric difference equation within the Hahn class, extending known relations to broader parameters.

Contribution

It introduces a qualitative, geometric method based on the q-Pearson equation, differing from traditional algebraic techniques, to extend orthogonality relations for q-polynomials.

Findings

Extended orthogonality relations for q-polynomials of the Hahn class.

Unified geometric framework for analyzing q-polynomial orthogonality.

Broadened parameter ranges for orthogonality of q-polynomials.

Abstract

The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the $q$-Pearson equation. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known $q$-polynomials of the Hahn class to a larger set of their parameters. A short version of this paper appeared in SIGMA 8 (2012), 042, 30 pages http://dx.doi.org/10.3842/SIGMA.2012.042.