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A generalization of the Heine-Stieltjes theorem
Department of Mathematics, Stockholm University.ORCID iD: 0000-0003-1055-1474
2011 (English)In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 34, no 1, 135-148 p.Article in journal (Refereed) Published
Abstract [en]

The Heine-Stieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.

Place, publisher, year, edition, pages
2011. Vol. 34, no 1, 135-148 p.
Keyword [en]
Heine-Stieltjes theorem, Heine-Stieltjes polynomials, Van Vleck polynomials, Hyperbolic polynomials, Real zeros, Interlacing zeros
National Category
URN: urn:nbn:se:kth:diva-78682DOI: 10.1007/s00365-010-9102-yISI: 000290805700006OAI: diva2:492761
QC 20120221Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2012-02-21Bibliographically approved

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Brändén, Petter
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