(1,1) - Coherent pairs on the unit circle

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 {()}≥0 and {()}≥0 satisfy [1] () + [1] −1() = () + −1(), ≠ 0, ≥ 1, where [1] () = +1()/( + 1). In this contribution, we consider the cases when U is the linear functional associated with the Lebesgue and Bernstein-Szeg˝o 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.The authors thank the referee the valuable comments. They greatly contributed to improve the contents of the paper. The work of Luis Garza was supported by Conacyt Grant no. 156668 and Beca Santander Iberoam´erica para J´ovenes Profesores e Investigadores (Mexico). The work of Francisco Marcell´an and Natalia C. Pinz´on-Cort´es has been supported by Direcci´on General de Investigaci´on, Desarrollo e Innovaci ´on, Ministerio de Econom´ıa y Competitividad of Spain, Grant MTM2012-36732-C03-01