transactions of the american mathematical society Volume 252, August 1979 ZEROS OF STIELTJES AND VANVLECKPOLYNOMIALS BY MAHFOOZ ALAM Abstract. The study of the polynomial solutions of the generalized Lamé differential equation gives rise to Stieltjes and Van Vleck polynomials. Marden has, under quite general conditions, established varied genera- lizations of the results proved earlier by Stieltjes, Van Vleck, Bôcher, Klein, and, Pólya, concerning the location of the zeros of such polynomials. We study the corresponding problem for yet another form of the generalized Lamé differential equation and generalize some recent results due to Zaheer and to Alam. Furthermore, applications of our results to the standard form of this differential equation immediately furnish the corresponding theorems of Marden. Consequently, our main theorem of this paper may be considered as the most general result obtained thus far in this direction. 1. Introduction. There exist [2] at most C(n + p — 2,p — 2) polynomials V(z) such that for <&(z) = V(z) the generalized Lamé differential equation dh/ dz2 + p a¡ y —-— dw dz *(z) m ■w= 0, (1.1) = i(z - aj) where <b(z) is a polynomial of degree at most (p — 2) and Oj, a, are complex constants, has a polynomial solution S(z) of degree n. Each polynomial V(z) and the corresponding polynomial S (z) are called [5, pp. 36-37] a Van Vleck polynomial and a Stieltjes polynomial, respectively (Marden [6, p. 934] calls V(z) the characteristic polynomial). We shall investigate the location of the zeros of the system of polynomials that arise in the study of the polynomial solutions of the differential equation d\f dz2 p i"/"1 , "j ) s«,, n(z-bj,)/n(z-ajs) f=i j=i + dw ' dz 4>(z) W¡. xWi=x(z-ajs) w = 0, (1.2) Received by the editors September 8, 1977 and, in revised form, June 19, 1978. AMS (MOS) subject classifications (1970). Primary 30A08; Secondary 33A70. Key words and phrases. Generalized Lamé differential equations, Stieltjes polynomials, Van Vleck polynomials, and covering functions. 'The results in this paper are partly contained in the author's doctoral thesis (1977) at Aligarh Muslim University, Aligarh, under the supervision of Dr. Neyamat Zaheer. The author wishes to thank him for his useful suggestions and advice all along. This work was performed under the research fellowship U.G.C. (Government of India). © 1979 American Mathematical Society 0002-9947/79/0000-0358/$03.00 197 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use