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Zbl 0828.42012
Magnus, Alphonse P.
Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. (English)
[J] J. Comput. Appl. Math. 57, No.1-2, 215-237 (1995). ISSN 0377-0427

This is a beautiful paper connecting deep mathematical results with problems in mathematical physics.\par The author restricts himself to semi-classical orthogonal polynomials (i.e., polynomials orthogonal -- may be on a set of arcs -- with respect to a weight function $w$ for which $w'/w$ is a rational function). The coefficients of the well-known three term recurrence relation satisfy nonlinear differential equations with respect to a certain parameter (here the work on monodromy by the Chudnovskij brothers plays a role (references [18]--[22] in the paper).\par The author treats five examples in varying detail:\par\hskip10mm 1. Jacobi weight with three factors $(1- x)^\alpha x^\beta(t- x)^\gamma$.\par\hskip10mm 2. $\exp(x^3/3+ tx)$ on $\{x: x^3< 0\}$.\par\hskip10mm 3. $\exp(- x^4- tx^2)$ on $\bbfR$ (simplest non-trivial Freud weight).\par\hskip10mm 4. $(x- t)^\rho \exp(- x^2)$ on $[t, \infty)$ ($t= \rho= 0$: Maxwell polynomials).\par\hskip10mm $\exp(- x^6- tx^2)$ on $\bbfR$.\par The paper contains an extensive list of references (93 items).
[M.G.de Bruin (Delft)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
34A34 Nonlinear ODE and systems, general

Keywords: Painlevé equation; semi-classical orthogonal polynomials; three term recurrence relation; nonlinear differential equations; Freud weight; Maxwell polynomials

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