A Generalized Freud Weight

TL;DR

This paper explores the connection between orthogonal polynomials with a generalized Freud weight and solutions of the fourth Painlevé equation, providing explicit formulas and differential equations for these polynomials.

Contribution

It establishes a link between recurrence coefficients of generalized Freud polynomials and Painlevé IV solutions, expressing coefficients via Wronskians of special functions.

Findings

Recurrence coefficients expressed in terms of Wronskians of parabolic cylinder functions.

Derived a second-order linear differential equation for the generalized Freud polynomials.

Established a differential-difference equation satisfied by these polynomials.

Abstract

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2\lambda+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda>-1$ and $t\in\mathbb{R}$, and classical solutions of the fourth Painlev\'{e} 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 Painlev\'{e} equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.