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On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential OperatorsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2007 (English)In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 149, no 2, 151-187 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2007. Vol. 149, no 2, 151-187 p.
##### Keyword [en]

Exactly-solvable operators, Asymptotic zero distribution, Root growth, Polynomial eigenfunctions, Eigenpolynomials
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-24129DOI: 10.1016/j.jat.2007.04.010ISI: 000252160800004OAI: oai:DiVA.org:su-24129DiVA: diva2:196823
#####

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Available from: 2007-03-09 Created: 2007-02-22 Last updated: 2011-02-16Bibliographically approved
##### In thesis

In this paper we study the class of differential operators with polynomial coefficients *Q*_{j} in one complex variable satisfying the condition deg*Q*_{j}*j* with equality for at least one *j*. We show that if deg*Q*_{k}<*k* then the root with the largest modulus of the *n*th degree eigenpolynomial *p*_{n} of *T* tends to infinity when *n*→∞, as opposed to the case when deg*Q*_{k}=*k*, which we have treated previously in [T. Bergkvist, H. RullgÅrd, 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.

1. Asymptotics of Eigenpolynomials of Exactly-Solvable Operators$(function(){PrimeFaces.cw("OverlayPanel","overlay196825",{id:"formSmash:j_idt1375:0:j_idt1379",widgetVar:"overlay196825",target:"formSmash:j_idt1375:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

doi
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