(1, 1)-Coherent Pairs on the Unit Circle. (10th November 2013) - Garza, Luis; Marcellán, Francisco; Pinzón-Cortés, Natalia C.
Abstract : A pair ( 𝒰, 𝒱 ) of Hermitian regular linear functionals on the unit circle is said to be a ( 1, 1 ) -coherent pair if their corresponding sequences of monic orthogonal polynomials { ϕ n ( x ) } n ≥ 0 and { ψ n ( x ) } n ≥ 0 satisfy ϕ n [ 1 ] ( z ) + a n ϕ n - 1 [ 1 ] ( z ) = ψ n ( z ) + b n ψ n - 1 ( z ), a n ≠ 0, n ≥ 1, where ϕ n [ 1 ] ( z ) = ϕ n + 1 ′ ( z ) / (n + 1) . In this contribution, we consider the cases when 𝒰 is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where 𝒱 is associated with either a positive nontrivial measure or its rational spectral transformation.