The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation

Abstract

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.

23 pages

References in corpus (8)

• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• The Hilbert series of N=1 SO(N_c) and Sp(N_c) SQCD, Painlevé VI and Integrable Systems
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
• Classical transcendental solutions of the Painlevé equations and their degeneration
• Painlevé V and the distribution function of discontinuous linear statistics in the Laguerre Unitary Ensembles
• The generalized Krawtchouk polynomials and the fifth Painlevé equation
• Perturbed Laguerre Unitary Ensembles, Painlevé V and Information Theory

Cited by in corpus (20)

• Properties of Generalized Freud Polynomials
• A Generalized Freud Weight
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
• On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials
• Orthogonal polynomials, Toda lattices and Painlevé equations
• A Generalised Sextic Freud Weight
• Generalised Airy Polynomials
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• On Integrable Ermakov-Painlevé IV Systems
• Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
• Large $z$ Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane
• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• An affine Weyl group characterization of polynomial Heisenberg algebras
• Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight
• Complete classification of rational solutions of $A_{2n}$-Painlevé systems
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
• Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV
• Painlevé V for a Jacobi unitary ensemble with random singularities