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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. (English) Zbl 0828.42012

This is a beautiful paper connecting deep mathematical results with problems in mathematical physics.

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).

The author treats five examples in varying detail:

       1. Jacobi weight with three factors (1-x) α x β (t-x) γ .

       2. exp(x 3 /3+tx) on {x:x 3 <0}.

       3. exp(-x 4 -tx 2 ) on (simplest non-trivial Freud weight).

       4. (x-t) ρ exp(-x 2 ) on [t,) (t=ρ=0: Maxwell polynomials).

       exp(-x 6 -tx 2 ) on .

The paper contains an extensive list of references (93 items).

42C05General theory of orthogonal functions and polynomials
34A34Nonlinear ODE and systems, general