### arXiv:1903.06159 [math-ph]AbstractReferencesReviewsResources

#### $q$-Racah ensemble and $q$-P$\left(E_7^{(1)}/A_{1}^{(1)}\right)$ Discrete Painléve equation

Published 2019-03-14Version 1

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 case of $q$-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 us a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

arXiv:math-ph/0203029 (Published 2002-03-18)

A new Lax pair for the sixth Painlevé equation associated with $\hat{\mathfrak{so}}(8)$

A Method for Constructing a Lax Pair for the Ernst Equation

Integrable Systems of Partial Differential Equations Determined by Structure Equations and Lax pair