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Zbl 0589.30004
Al-Rashed, Abdallah M.; Zaheer, Neyamat
Zeros of Stieltjes and Van Vleck polynomials and applications. (English)
[J] J. Math. Anal. Appl. 110, 327-339 (1985). ISSN 0022-247X

The prime interest is in locating the zeros of the system of polynomials that arise in the study of the polynomial solutions of the generalized Lamé's differential equation (GLDE) $$ \{D\sp 2\sb z+(\sum\sp{p}\sb{j=1}\alpha\sb j/(z-a\sb j))D\sb z+\Phi (z)/\prod\sp{p}\sb{j=1}(z-\quad a\sb j)\}w(z)=0 $$ where $\Phi$ (z) is a polynomial of degree at most p-2 (p$\ge 2)$ and $\alpha\sb j$ and $a\sb j$ are complex constants. It is known that there are at most $C(n+p-2,p- 2)$ Van Vleck polynomials $V(z):=\Phi (z)$ such that the GLDE has a Stieltjes polynomial solution of degree n. The main theorem develops the notion of reflector curves and sets to locate the zeros of these polynomials relative to a prescribed location of the constants. Several classical theorems follow as special cases. Applications to problems arising in physics, fluid mechanics and the location of the complex zeros of Jacobi polynomials are discussed.
[Peter McCoy]

MSC 2000:: *30C10 Polynomials (one complex variable)
30C15 Zeros of polynomials, etc. (one complex variable)
33C45 Orthogonal polynomials and functions of hypergeometric type
34A05 Methods of solution of ODE

Keywords: Lamé's differential equation; Van Vleck polynomials; Stieltjes polynomial; zeros; Jacobi polynomials

Cited in: Zbl 0618.30007

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