Some properties of Jacobi polynomials orthogonal on a circle

Let $\{\psi^{(\alpha,\beta)}_n(z)\}_{n=0}^\infty$ be a system of Jacobi polynomials that is orthonormal on the circle $|z|=1$ with respect to the weight $(1-\cos\tau)^{\alpha+1/2}(1+\cos\tau)^{\beta+1/2}$ ($\alpha,\beta>-1$), and let $\psi_n^{(\alpha,\beta)*}(z):=z^n\overline{\psi_n^{(\alpha,\beta)}(1/\overline z)}$. We establish relations between the polynomial $\psi_n^{(\alpha,-1/2)}(z)$ and the $n$-th $(C,\alpha-1/2)$-mean of the Maclaurin series for the function $(1-z)^{-\alpha-3/2}$ and also between the polynomial $\psi_n^{(\alpha,-1/2)*}(z)$ and the $n$-th $(C,\alpha+1/2)$-mean of the Maclaurin series for the function $(1-z)^{-\alpha-1/2}$. We use these relations to derive an asymptotic formula for $\psi_n^{(\alpha, -1/2)}(z)$; the formula is uniform inside the disk $|z|<1$. It follows that $\psi_n^{(\alpha,-1/2)}(z)\neq0$ in the disk $|z|\le\rho$ for fixed $\rho\in(0,1)$ and $\alpha>-1$ if $n$ is sufficiently large.