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arXiv · 2110.13420

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

Peter J Forrester·Shi-Hao Li·Bo-Jian Shen·Guo-Fu Yu
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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.

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Peter J Forrester, Shi-Hao Li, Bo-Jian Shen, Guo-Fu Yu. 2021-10-26. $q$-Pearson pair and moments in $q$-deformed ensembles. https://arxiv.org/abs/2110.13420

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