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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the zeros of Stieltjes and Van Vleck polynomials


Authors: Neyamat Zaheer and Mahfooz Alam
Journal: Trans. Amer. Math. Soc. 229 (1977), 279-288
MSC: Primary 30A08
MathSciNet review: 0435367
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Abstract: Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas' theorem to systems of partial fraction sums.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0435367-3
PII: S 0002-9947(1977)0435367-3
Keywords: Generalized Lamé differential equations, Stieltjes polynomials, Van Vleck polynomials
Article copyright: © Copyright 1977 American Mathematical Society