The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlev\'e equation

TL;DR

This paper demonstrates that recurrence coefficients of semi-classical Laguerre polynomials satisfy the fourth Painlevé equation, connecting orthogonal polynomial theory with integrable systems through multiple analytical approaches.

Contribution

It establishes the link between recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation, using ladder operators, isomonodromy deformations, and Toda systems.

Findings

Recurrence coefficients satisfy the fourth Painlevé equation as functions of a parameter.

Different analytical methods yield consistent results.

A relation between Freud and semi-classical Laguerre weights is derived from Bäcklund transformations.

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 Painlev\'e 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 B\"acklund transformation of the fourth Painlev\'e equation.