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

 

Zeros of Stieltjes and Van Vleck polynomials


Author: Mahfooz Alam
Journal: Trans. Amer. Math. Soc. 252 (1979), 197-204
MSC: Primary 30C15
MathSciNet review: 534117
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Abstract: The study of the polynomial solutions of the generalized Lamé differential equation gives rise to Stieltjes and Van Vleck polynomials. Marden has, under quite general conditions, established varied generalizations of the results proved earlier by Stieltjes, Van Vleck, Bocher, Klein, and, Pólya, concerning the location of the zeros of such polynomials. We study the corresponding problem for yet another form of the generalized Lamé differential equation and generalize some recent results due to Zaheer and to Alam. Furthermore, applications of our results to the standard form of this differential equation immediately furnish the corresponding theorems of Marden. Consequently, our main theorem of this paper may be considered as the most general result obtained thus far in this direction.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0534117-1
PII: S 0002-9947(1979)0534117-1
Keywords: Generalized Lamé differential equations, Stieltjes polynomials, Van Vleck polynomials, and covering functions
Article copyright: © Copyright 1979 American Mathematical Society