On the zeros of orthogonal polynomials: the elliptic case (Q1885373)

Let \(E=[-1,\alpha]\cup[\beta,1],\;-1<\alpha<\beta<1,\) and let \((p_n)\) be orthogonal on \(E\) with respect to the weight function \(((1-x^2)(x-\alpha)(x-\beta))^{\pm 1/2}/W(x),\) where \(W\) is positive on \(E\) and \(W\in C^3(E).\) The author gives the exact number (in terms of elliptic functions) of zeros of \(p_n\) in the two intervals and in the gap \((\alpha,\beta).\) Also, the accumulation points of the zeros are completely described and corresponding results are proved if \(W\) has a different sign on two intervals of \(E.\)