$q$-Pearson pair and moments in $q$-deformed ensembles

TL;DR

This paper explores moments in $q$-deformed ensembles, linking orthogonal polynomials, hypergeometric functions, and $q$-difference equations to advance understanding of spectral properties in $q$-lattice models.

Contribution

It introduces a systematic study of moments in $q$-ensembles using hypergeometric functions and derives a fourth order $q$-difference equation generalizing classical results.

Findings

Expression of density moments via ${}_3 $ hypergeometric polynomial

Derivation of a fourth order $q$-difference equation for the $q$-Laplace transform

Extension of classical spectral analysis to $q$-deformed ensembles

Abstract

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the $q$-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little $q$-Laguerre weight, a particular ${}_3 \phi_2$ basic hypergeometric polynomial is used to express density moments. The second approach is to study the $q$-Laplace transform of the un-normalised measure. Using integrability properties associated with the $q$-Pearson equation for the $q$-classical weights, a fourth order $q$-difference equation is obtained, generalising a result of Ledoux in the continuous classical cases.