Preud's equations for orthogonal polynomials as discrete Painlev\'e equations
Classical Analysis and ODEs
2016-09-06 v1
Abstract
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.
Keywords
Cite
@article{arxiv.math/9611218,
title = {Preud's equations for orthogonal polynomials as discrete Painlev\'e equations},
author = {Alphonse P. Magnus},
journal= {arXiv preprint arXiv:math/9611218},
year = {2016}
}