The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
type: Item
instance of: scholarly article
title: The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
zbMATH Open document ID: 1253.33012
DOI: 10.1088/1751-8113/45/20/205201
author: Walter Van Assche
author: Lun Zhang
author: Galina Filipuk
published in: Journal of Physics A: Mathematical and Theoretical
publication date: 2012-06-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.5229
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 6045423
zbMATH Keywords: semi-classical orthogonal polynomials
zbMATH Keywords: recurrence coefficients
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Bäcklund transformations
MaRDI profile type: Publication
OpenAlex ID: W1994029617
arXiv ID: 1105.5229
review text: For the monic polynomials \(p_n(x)\) orthogonal on the half-line with respect to the weight \(w(x)=x^{\alpha}e^{-x^2+tx}\) depending on the constant \(\alpha>-1\) and a real parameter \(t\), the authors prove that the coefficient \(b_n\) in the three-term recurrence relation \(xp_n=a_np_{n+1}+b_np_n+a_np_{n-1}\), as a function of \(t\), satisfies the classical fourth Painlevé equation PIV with the parameters determined by \(\alpha\) and \(n\). The authors present several proofs, one of which is based on the use of the differential-difference system for the recurrence coefficients \(a_n,b_n\), another one uses the ladder operator approach, and the last one utilizes the Lax pair for the orthogonal polynomials. Finally, the authors explain the relation between the mentioned above differential-difference system for the recurrence coefficients and the Bäcklund transformations of PIV.
reviewed by: Andrei A. Kapaev
description: scientific article; zbMATH DE number 6045423