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

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)(3) + n), and a(n)(2) satisfies a Painleve: P-IV equation.

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