Preud's equations for orthogonal polynomials as discrete Painlevé equations

Details Preud's equations for orthogonal polynomials as discrete Painlevé equations Preud's equations for orthogonal polynomials as discrete Painlevé equations

1996-11-29

arxiv.org/abs/math/9611218v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2tw_n + w_n(w_n+1 + w_n + w_n-1) for the recurrence coefficients p_n+1 = xp_n - w_np_n-1 of the orthogonal polynomials related to the weight w(x) = exp(-4(tx^3 + x^4)) (notation of [26, pp. 34-36]). This example appears in practically all the references below. The connection with discrete Painlev\'e equations is described here.

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