The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

Galina Filipuk, Walter Van Assche, Lun Zhang
2012 Journal of Physics A: Mathematical and Theoretical  
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
more » ... tion. 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.
doi:10.1088/1751-8113/45/20/205201 fatcat:dusrvnaa35eldfivmre7meqxcm