Painlev 'e-type differential equations for the recurrence coefficients\n of semi-classical orthogonal polynomials.

1993/07/08 by Alphonse Magnus, Magnus, Alphonse P. · 5 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.math/9307218

openalex publication_date 1993/07/08 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal\npolynomials related to a weight function w such that w'/w is a rational\nfunction) are shown to be solutions of non linear differential equations with\nrespect to a well-chosen parameter, according to principles established by D.\nG. Chudnovsky. Examples are given. For instance, the recurrence coefficients in\na_n+1p_n+1(x)=xp_n(x) -a_np_n-1(x) of the orthogonal polynomials\nrelated to the weight \exp(-x⁴/4-tx²) on blackb R / satisfy 4a_n³ na_n = (3a_n⁴+2ta_n²-n)(a_n⁴+2ta_n²+n), and a_n² satisfies a Painlev 'e\n rm P _{rm IV} equation.\n

Cited by

• Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the D₅ Sakai surface
• Recurrence coefficients for the semiclassical Laguerrre weight and d-P(A₂⁽¹⁾/E₆⁽¹⁾) equations
• A class of Truncated Freud polynomials
• Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials
• Differential equations for a class of semiclassical orthogonal polynomials on the unit circle

Related

• Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach
• Preud's equations for orthogonal polynomials as discrete Painlevé equations
• q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials
• The generalized Krawtchouk polynomials and the fifth Painlevé equation
• Painlevé equations for semi-classical recurrence coefficients