Further Results on Stieltjes and Van Vleck Polynomials PDF

Keywords

polynomials, Lame's differential equations, Stieltjes, Van Vleck

Abstract

Stieltjes and Van Vleck polynomials are the polynomial solutions of the generalized Lame's differential equation. The problem of determining the relative location of the zeros of such polynomials and the complex constants occurring in the said differential equation has recently been studied by Al-Rashed and Zaheer (1985) under quite general conditions by introducing the concept of reflector regions. The present study solves the corresponding problem for yet another form of the generalized Lame's differential equation and offers much more general versions of some results due to Zaheer and Alam. Furthermore, when applied to the standard form of the generalized Lame's differential equation, our results in this paper deduce the corresponding results to Al-Rashed and Zaheer (1985), including some due to Mardan, Bocher, Klein, Polya, Stieltjes and Van Vleck as special cases