2011/05/26 by Galina Filipuk, Walter Van Assche, Lun Zhang · 1 citation
Mathematics · #math.CA #msc:33C47 #msc:33E17 #msc:34M55
paper · pdf · doi:10.1088/1751-8113/45/20/205201
published as J. Phys. A: Math. Theor. 45 (2012), 205201 · 18 pages
arxiv created 2011/05/26 · arxiv updated 2013/10/04
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 Painlevé 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 Bäcklund transformation of the fourth Painlevé equation.