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

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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 nonlinear differential equations with respect to a well-chosen parameter, according to principles established by D. Chudnovsky and G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in a n + 1 P n + 1 ( x ) = xp n ( x ) − a n p n − 1 ( x ) of the orthogonal polynomials related to the weight exp ( − x 4 4 − tx 2 ) on R satisfy 4a n 3 a ̈ n = (3a n 4 + 2ta n 2 − n)(a n 4 + 2ta n 2 + n) , and a n 2 satisfies a Painlevé P IV equation.