Generalized Coherent Pairs on the Unit Circle and Sobolev Orthogonal Polynomials

A pair of regular Hermitian linear functionals (U ,V) is said to be an (M,N)-coherent pair of order m on the unit circle if their corresponding sequences of monic orthogonal polynomials {φn(z)}n>0 and {ψn(z)}n>0 satisfy M ∑ i=0 ai,nφ (m) n+m−i(z) = N ∑ j=0 bj,nψn−j(z), n > 0, where M,N,m > 0, ai,n and bj,n, for 0 6 i 6 M , 0 6 j 6 N , n > 0, are complex numbers such that aM,n 6= 0, n > M , bN,n 6= 0, n > N , and ai,n = bi,n = 0, i > n. When m = 1, (U ,V) is called a (M,N)-coherent pair on the unit circle. We focus our attention on the Sobolev inner product 〈 p(z), q(z) 〉 λ = 〈 U , p(z)q(1/z) 〉 + λ 〈 V , p(z)q(m)(1/z) 〉 , λ > 0, m ∈ Z, assuming that U and V is an (M,N)-coherent pair of orderm on the unit circle. We generalize and extend several recent results of the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. Besides, we analyze the cases (M,N) = (1, 1) and (M,N) = (1, 0) in detail. In particular, we illustrate the situation when U is the Lebesgue linear functional and V is the Bernstein–Szegő linear functional. Finally, a matrix interpretation of (M,N)-coherence is given.