(1,1)‎-Coherent Pairs on the Unit Circle

Abstract EN

A pair (?,?) 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 {ϕn(x)}n≥0 and {ψn(x)}n≥0 satisfy ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z), an≠0, n≥1, where ϕn[1](z)=ϕn+1′(z)/(n+1).

In this contribution, we consider the cases when ? is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where ? is associated with either a positive nontrivial measure or its rational spectral transformation.