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) = n k=0 ϕ 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 class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.

Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions

“…with respect to the measure µ (b) given above. The polynomials R n (b; •) and the associated orthogonal polynomials Φ n (b; •), in addition to have been studied in [31], they have been used as examples in a sequence of papers [5,7,8,10,12,18,19], without knowing anything about their connection to the CRR polynomials. The results obtained in [18] are focused on the three term recurrence of the type (1.7) and the associated generalized eigenvalue problem (with these respect see also [17] and [40]).…”