On the zeros of Stieltjes and Van Vleck polynomials

Abstract:

Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas’ theorem to systems of partial fraction sums.