A Generalized Freud Weight

Abstract

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2λ+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda >-1$ and $t\in \mathbb{R}$, 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 that arise in the description of special function solutions of the fourth Painlevé equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.

22 pages, Studies in Applied Mathematics, accepted for publication

References in corpus (1)

• Painleve IV and degenerate Gaussian Unitary Ensembles

Cited by in corpus (12)

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• A Generalised Sextic Freud Weight
• Generalised Airy Polynomials
• The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation
• On Integrable Ermakov-Painlevé IV Systems
• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• On Freud-Sobolev type orthogonal polynomials
• Generalised higher-order Freud weights
• Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight
• Symmetric Sextic Freud Weight