Christoffel formula for kernel polynomials on the unit circle

Abstract

Given a nontrivial positive measure $μ$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;μ) = \sum_{k=0}^{n}\overline{φ_{k}(w;μ)}\,φ_{k}(z;μ)$, $n \geq 0$, where $φ_{k}(\cdot; μ)$ are the orthonormal polynomials with respect to the measure $μ$. Let the positive measure $ν$ on the unit circle be given by $d ν(z) = |G_{2m}(z)|\, d μ(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing $\{K_n(z,w;ν)\}_{n \geq 0}$ directly in terms of $\{K_n(z,w;μ)\}_{n \geq 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;μ) $ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $ \{c_n(μ)\}_{n=1}^{\infty}$ and $ \{g_{n}(μ)\}_{n=1}^{\infty}$, with $0<g_n<1 $ for $n\geq 1$. The double sequence $\{(c_n(μ), g_{n}(μ))\}_{n=1}^{\infty}$ characterizes the measure $μ$. A natural question about the relation between the parameters $c_n(μ)$, $g_n(μ)$, associated with $μ$, and the sequences $c_n(ν)$, $g_n(ν)$, corresponding to $ν$, 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.