Zeros of polynomials orthogonal on two arcs of the unit circle

Abstract

In this paper we study polynomials $(P_n)$ which are hermitian orthogonal on two arcs of the unit circle with respect to weight functions which have square root singularities at the end points of the arcs, an arbitrary nonvanishing trigonometric polynomial $\mathcal{A}$ in the denominator and possible point measures at the zeros of $\mathcal{A}$. First we give an explicit representation of the orthogonal polynomials $P_n$ in terms of elliptic functions. With the help of this representation for sufficiently large $n$ the number of zeros of $P_n$ which are in an $ε$-neighbourhood of each of the arcs are determined. Finally it is shown that the accumulation points of the zeros of $(P_n)$ which are not attracted to the support lie on a Jordan arc running within the unit disk from one of the arcs to the other one. The accumulation points lie dense on the Jordan arc if the harmonic measures of the arcs are irrational. If the harmonic measures are rational then there is only a finite set of accumulation points on the Jordan arc.