The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

Abstract

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Bäcklund transformation of the fourth Painlevé equation.

18 pages

References in corpus (4)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Random Matrix Theory and the Sixth Painlevé Equation
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• Generating Function Associated with the Determinant Formula for the Solutions of the Painleve' II Equation

Cited by in corpus (21)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble
• A Generalized Freud Weight
• Properties of Generalized Freud Polynomials
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials
• Ladder operators and differential equations for multiple orthogonal polynomials
• On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight
• Generalised Airy Polynomials
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• On Freud-Sobolev type orthogonal polynomials
• Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
• Painlevé IV, $\sigma -$Form and the Deformed Hermite Unitary Ensembles
• Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight
• Symmetric Sextic Freud Weight
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• On properties of a deformed Freud weight
• Painlevé V for a Jacobi unitary ensemble with random singularities