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Zeros of Stieltjes and Van Vleck polynomials and applications. (English) Zbl 0589.30004
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) \[ \{D^ 2_ z+(\sum^{p}_{j=1}\alpha_ j/(z-a_ j))D_ z+\Phi (z)/\prod^{p}_{j=1}(z-\quad a_ j)\}w(z)=0 \] where \(\Phi\) (z) is a polynomial of degree at most p-2 (p\(\geq 2)\) and \(\alpha_ j\) and \(a_ j\) 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.
Reviewer: Peter McCoy

MSC:
30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34A05 Explicit solutions, first integrals of ordinary differential equations
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