some Bar-le-Duc stuff: manuscript of Nov. 1983 (pdf)   summary 1 (pdf)   summary 2 (pdf)   summary 3 (pdf)
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## A proof of Freud's conjecture about the orthogonal polynomials related to |x|ρexp(-x2m), for integer m.

 Book Series Lecture Notes in Mathematics Éditeur Springer Berlin / Heidelberg ISSN 0075-8434 (Print) 1617-9692 (Online) Volume Volume 1171/1985 Book Polynômes Orthogonaux et Applications DOI 10.1007/BFb0076527 Copyright 1985 ISBN 978-3-540-16059-5 Category II. Conferenciers Ou Contributeurs DOI 10.1007/BFb0076565 Pages 362-372 Subject Collection
A proof of Freud's conjecture about the orthogonal polynomials related to |x|ρexp(−x2m), for integer m.

Alphonse P. Magnus1

 (1) Institut de Mathématique, Université Catholique de Louvain, chemin du Cyclotron, 2, 1348 Louvain-la-neuve, Belgium
Abstract
Let anpn(x)=xpn−1(x)−an−1pn−2(x) be the recurrence relation of the orthonormal polynomials related to the weight function |x|ρ exp(−|x|α), ρ < −1, α < 0, on the whole real line. Freud's conjecture states that The proof for an even integer α=2m uses nonlinear equations Fn(a)=n + ρ odd(n), considered by Freud himself. It is shown that Fn(a*)−n=o(n) when n → ∞, where a*n is the expected asymptotically valid estimate [n/C(α)]1/α. Bounds on an−a*n are obtained through the invertibility properties of the matrix [ak ∂Fn (a)/∂ak], shown to be symmetric and positive definite. The numerical computation of the solution by Newton's method is considered.