An Extension of the Coherent Pair of Measures of the Second Kind on the Unit Circle - Garza, Lino G.; Marcellán, F.; Ranga, A. Sri [UNESP] (Springer Nature, 2021)
This paper deals with sequences of monic polynomials { Φn(μk;z)}n≥0, k = 0, 1, orthogonal with respect to two nontrivial Borel measures μk, k = 0, 1, supported on the unit circle, satisfying (formula presented) n ≥ 3, where bn ≠ 0. We find examples of pairs of measures (μ0, μ1) for which this property holds. The analysis of polynomials orthogonal with respect to the Sobolev inner product associated with the pair of measures (μ0, μ1) is presented. Some properties concerning their connection coefficients are given.
Keywords: Coherent pairs of measures of the second kind; Hessenberg matrices; Orthogonal polynomials on the unit circle; Probability measures on the unit circle; Sobolev inner products on the unit circle