Christoffel formula for kernel polynomials on the unit circle

Given a nontrivial positive measure μ on the unit circle T, the associated Christoffel–Darboux kernels are K_n(z,w;μ)=∑_k=0 ⁿφ_k(w;μ)¯φ_k(z;μ), n≥0, where φ_k(⋅;μ) are the orthonormal polynomials with respect to the measure μ. Let the positive measure ν on the unit circle be given by dν(z)=|G_2m(z)|dμ(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≥0 directly in terms of {K_n(z,w;μ)}_n≥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 ^∞ and {g_n(μ)}_n=1 ^∞, with 0<g_n<1 for n≥1. The double sequence {(c_n(μ),g_n(μ))}_n=1 ^∞ 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 T), a measure for which the Christoffel–Darboux kernels, with w=1, are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel–Darboux kernels, again with w=1, are given by hypergeometric polynomials.