Zeros of Stieltjes and Van Vleck polynomials and applications (Q1073944)

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.