On the Zeros of Orthogonal Polynomials: The Elliptic Case

Abstract

Let E = [–1, α] \cup [β, 1], –1 < α < β < 1, and let (pn) be orthogonal on E with respect to the weight function ((1 – x2)(x – α)(x – β))±1/2/W(x), where W is positive on E and W \in C3(E). Although A. Markoff has already studied the behavior of the zeros of such orthogonal polynomials, surprisingly, most of the basic questions such as: How many zeros has pn in each of the two intervals? When does there appear a zero of pn in the gap [α, β]? Are the accumulation points of the zeros of (pn) dense in the gap, . . . ?; remain open so far. In this paper we answer these questions with the help of elliptic functions. More precisely, we give the exact number of zeros of pn in the two intervals, and we show that every point of (α, β) is an accumulation point of (pn) if E is not the inverse image of [–1, 1] under a polynomial mapping and that there is a finite number of accumulation points otherwise. Also, corresponding results are proved if W has a different sign on the two intervals of E.