Christoffel formula for kernel polynomials on the unit circle

Abstract: Given a nontrivial positive measure $\mu$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;\mu) = \sum_{k=0}^{n}\overline{\varphi_{k}(w;\mu)}\,\varphi_{k}(z;\mu)$, $n \geq 0$, where $\varphi_{k}(\cdot; \mu)$ are the orthonormal polynomials with respect to the measure $\mu$. Let the positive measure $\nu$ on the unit circle be given by $d \nu(z) = |G_{2m}(z)|\, d \mu(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing ${K_n(z,w;\nu)}{n \geq 0}$ directly in terms of ${K_n(z,w;\mu)}{n \geq 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;\mu) $ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $ {c_n(\mu)}{n=1}^{\infty}$ and $ {g{n}(\mu)}{n=1}^{\infty}$, with $0<g_n<1 $ for $n\geq 1$. The double sequence ${(c_n(\mu), g{n}(\mu))}_{n=1}^{\infty}$ characterizes the measure $\mu$. A natural question about the relation between the parameters $c_n(\mu)$, $g_n(\mu)$, associated with $\mu$, and the sequences $c_n(\nu)$, $g_n(\nu)$, corresponding to $\nu$, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of the unit circle), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.