Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations

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 Painlevé equations.

48 pages, 2 figures

References in corpus (2)

• What is $\dots$ a multiple orthogonal polynomial?
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Cited by in corpus (9)

• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Multiple orthogonal polynomials: Pearson equations and Christoffel formulas
• Orthogonal polynomials, Toda lattices and Painlevé equations
• Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials
• Asymptotics of Discrete $q$-Freud $\mathrm{II}$ orthogonal polynomials from the $q$-Riemann Hilbert Problem
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• Multiple Orthogonal Polynomials of two real variables
• Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths
• Discrete equations from Bäcklund transformations of the fifth Painlevé equation