GENERALIZED COHERENT PAIRS ON THE UNIT CIRCLE AND SOBOLEV ORTHOGONAL POLYNOMIALS

Francisco Marcellán, Natalia C. Pinzón-Cortés

A pair of regular Hermitian linear functionals $(\U,\V)$ is said to be an \emph{$(M,N)$-coherent pair of order $m$ on the unit circle} if their corresponding sequences of monic orthogonal polynomials $\{\phi_n(z)\}_{n\geq0}$ and $\{\psi_n(z)\}_{n\geq0}$ satisfy $$um_{i=0}^M a_{i,n} hi^{(m)}_{n+m-i}(z)=um_{j=0}^N b_{j,n} si_{n-j}(z), \quad n\geq 0,$$ where $M,N,m\geq0$, $a_{i,n}$ and $b_{j,n}$, for $0\leq i\leq M$, $0\leq j\leq N$, $n\geq0$, are complex numbers such that $a_{M,n}\neq0$, $n\geq M$, $b_{N,n}\neq0$, $n\geq N$, and $a_{i,n}=b_{i,n}=0$, $i>n$. When $m=1$, $(\U,\V)$ is called a \emph{$(M,N)$-coherent pair on the unit circle}. We focus our attention on the Sobolev inner product $$\biglangle p(z),q(z)\bigr\rangle_ambda=\biglangle\U, p(z)verline{q}(1/z) \bigr\rangle +\biglangle\V, p^{(m)}(z)verline{q^{(m)}}(1/z)\bigr\rangle, \quad ambda>0,\,mı\Z^+,$$ assuming that $\U$ and $\V$ is an $(M,N)$-coherent pair of order $m$ 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.