First and second kind paraorthogonal polynomials and their zeros

Given a probability measure μ with infinite support on the unit circle ∂D = {z : |z| = 1}, we consider a sequence of paraorthogonal polynomials hn(z, λ) vanishing at z = λ where λ ∈ ∂D is fixed. We prove that for any fixed z0 6∈ supp(dμ) distinct from λ, we can find an explicit ρ > 0 independent of n such that either hn or hn+1 (or both) has no zero inside the disk B(z0, ρ), with the possible exception of λ. Then we introduce paraorthogonal polynomials of the second kind, denoted sn(z, λ). We prove three results concerning sn and hn. First, we prove that zeros of sn and hn interlace. Second, for z0 an isolated point in supp(dμ), we find an explicit radius ρ̃ such that either sn or sn+1 (or both) have no zeros inside B(z0, ρ̃). Finally we prove that for such z0 we can find an explicit radius such that either hn or hn+1 (or both) has at most one zero inside the ball B(z0, ρ̃).