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arXiv · 1903.06159

$q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlev\'e equation

Anton Dzhamay·Alisa Knizel
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Abstract

The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlev\'e equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

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Anton Dzhamay, Alisa Knizel. 2019-03-14. $q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ discrete Painlev\'e equation. https://arxiv.org/abs/1903.06159

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