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Zeros of Stieltjes and Van Vleck Polynomials

Mahfooz Alam
Transactions of the American Mathematical Society
Vol. 252 (Aug., 1979), pp. 197-204
DOI: 10.2307/1998084
Stable URL: http://www.jstor.org/stable/1998084
Page Count: 8
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Zeros of Stieltjes and Van Vleck Polynomials
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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, Bôcher, 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 maintheorem of this paper may be considered as the most general result obtained thusfar in this direction.

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