The relationship between semi-classical Laguerre polynomials and the fourth Painlev\'e equation

TL;DR

This paper explores how recurrence coefficients of semi-classical Laguerre polynomials relate to solutions of the fourth Painlevé equation, linking orthogonal polynomials with special function solutions of a nonlinear differential equation.

Contribution

It establishes a connection between orthogonal polynomial recurrence coefficients and classical solutions of the fourth Painlevé equation through Wronskians of parabolic cylinder functions.

Findings

Recurrence coefficients can be expressed via Wronskians of parabolic cylinder functions.

The study links orthogonal polynomials with Painlevé equations.

Provides explicit formulas for coefficients in terms of special functions.

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 Painlev\'e 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 Painlev\'e equation.