Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.

Details Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.

1993-07-09

arxiv.org/abs/math/9307218v1

Mathematics

Classical Analysis and ODEs

Scientific paper

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ satisfies a Painlev\'e ${\rm P}_{\rm IV}$ equation.

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