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

## TL;DR

This paper links the $q$-Racah polynomial ensemble to the discrete $q$-Painlevé equation $q$-P$(E_7^{(1)}/A_{1}^{(1)})$, providing new insights and a novel Lax pair with symmetry.

## Contribution

It establishes a connection between $q$-Racah ensembles and a specific discrete Painlevé equation, including a new Lax pair with symmetry.

## Key findings

- Expressed gap probabilities via solutions to the $q$-Painlevé equation.
- Derived a new Lax pair with involutive symmetry.
- Enhanced understanding of the relationship between orthogonal polynomials and Painlevé equations.

## 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.