The Symmetrical $H_{q}$-Semiclassical Orthogonal Polynomials of Class One

TL;DR

This paper classifies symmetrical $H_q$-semiclassical orthogonal polynomials of class one using quadratic decomposition and duality, providing explicit recurrence coefficients, distributional equations, moments, and representations.

Contribution

It introduces a comprehensive classification of these polynomials, detailing their properties and explicit formulas, which advances understanding of $H_q$-semiclassical orthogonal polynomials.

Findings

Explicit recurrence coefficients derived

Distributional equations of Pearson type established

Moments and integral/discrete representations provided

Abstract

We investigate the quadratic decomposition and duality to classify symmetrical $H_{q}$-semiclassical orthogonal $q$-polynomials of class one where $H_{q}$ is the Hahn's operator. For any canonical situation, the recurrence coefficients, the $q$-analog of the distributional equation of Pearson type, the moments and integral or discrete representations are given.