# Orthogonal and multiple orthogonal polynomials, random matrices, and Painlev\'e equations

## TL;DR

This paper introduces orthogonal and multiple orthogonal polynomials, explores their applications in random matrix theory, and discusses their connection to Painlevé equations, highlighting their significance in mathematical physics and related fields.

## Contribution

It provides an overview of the theory of orthogonal and multiple orthogonal polynomials and elucidates their links with Painlevé equations in the context of random matrices.

## Key findings

- Orthogonal polynomials are fundamental in mathematical physics and probability.
- Multiple orthogonal polynomials extend classical theory with new applications.
- Connections between orthogonal polynomials and Painlevé equations are established.

## Abstract

Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics and probability and many other disciplines. In these notes we give an introduction to the use of orthogonal polynomials in random matrix theory, we explain the notion of multiple orthogonal polynomials, and we show the link with certain non-linear difference and differential equations known as Painlev\'e equations.