On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

Authors:
Tanja Bergkvist
Affiliations:
Department of Mathematics, University of Stockholm, S10691 Stockholm, Sweden
Venue:
Journal of Approximation Theory
Year:
2007

Citing 6
Cited 1

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Orthogonal polynomial solutions of linear ordinary differential equations

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On a class of equilibrium problems in the real axis

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Abstract

In this paper we study the class of differential operators T=@?"j"="1^kQ"jD^j with polynomial coefficients Q"j in one complex variable satisfying the condition degQ"j=~, as opposed to the case when degQ"k=k, which we have treated previously in [T. Bergkvist, H. RullgArd, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153-171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of p"n. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.