Zeros of Polynomials Orthogonal on Several Intervals

Let a1 < a2 < . . . < a2l, Ej = [a2j−1, a2j ], put E = ⋃l j=1 Ej and H(x) = ∏2l j=1(x − aj). Furthermore let pn(x) = x + . . . be the polynomial of degree n orthogonal on E with respect to a weight function of the form w/ √ −H with square root singularities at the boundary points of E and w ∈ C(E). We study and answer the following questions: how many zeros has pn in the interval Ej , j ∈ {1, . . . , l}, when does there appear a zero of pn in a gap (a2j , a2j+1), are the accumulation points of the zeros of (pn) dense in the gaps, . . . . First we give the number of zeros of pn in the interval Ej , j = 1, . . . , l, in terms of the harmonic measure of Ej and the logarithmic mean value of the weight function. Then a criteria for the appearance of a zero of pn respectively of an accumulation point of zeros of (pn) in a gap is derived. As a consequence we obtain that every point of [a1, a2l] \ E is an accumulation point of zeros of (pn) if the harmonic measures of the intervals are linearly independent over the rationals. If the harmonic measures are rational then there is a finite number of accumulation points of zeros of (pn) in [a1, a2l] \ E only.