First and second kind paraorthogonal polynomials and their zeros

Abstract

Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We prove that for any fixed $z_0 \not \in \supp(d\mu)$ distinct from $\lambda$, we can find an explicit $\rho>0$ independent of $n$ such that either $\h_n$ or $\h_{n+1}$ (or both) has no zero inside the disk $B(z_0, \rho)$, with the possible exception of $\lambda$. Then we introduce paraorthogonal polynomials of the second kind, denoted $\s_n(z,\lambda)$. We prove three results concerning $\s_n$ and $\h_n$. First, we prove that zeros of $\s_n$ and $\h_n$ interlace. Second, for $z_0$ an isolated point in $\supp(d\mu)$, we find an explicit radius $\rt$ such that either $\s_n$ or $\s_{n+1}$ (or both) have no zeros inside $B(z_0,\rt)$. Finally we prove that for such $z_0$ we can find an explicit radius such that either $\h_n$ or $\h_{n+1}$ (or both) has at most one zero inside the ball $B(z_0,\rt)$.