q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials

2008/08/07 by Lies Boelen, Boelen, Lies, Christophe Smet +3
Computer Science · Mathematics · Physics and Astronomy · #33D45 #34M55 (Primary) 65Q05 #39A13 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0808.0982

openalex publication_date 2008/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an asymmetric q-Painlevé equation. We will derive this using q-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this q-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution which gives the recurrence coefficients. We establish a connection with α-q-P_V.

Citations

• A differential equation for orthogonal polynomials

Related

• Preud's equations for orthogonal polynomials as discrete Painlevé equations
• Painlev 'e-type differential equations for the recurrence coefficients\n of semi-classical orthogonal polynomials.
• The generalized Krawtchouk polynomials and the fifth Painlevé equation
• Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach
• Non linear difference equations arising from a deformation of the q-Laguerre weight