(M, N)-Coherent pairs of linear functionals and Jacobi matrices

摘要

A pair of regular linear functionals (U, V) in the linear space of polynomials with complex coefficients is said to be an (M, N)-coherent pair of order m if their corresponding sequences of monic orthogonal polynomials {P_n(x)}_(n≥0) and {Q_n(x)}_(n≥0) satisfy a structure relation ∑_(i=0)~Ma_(i, n) P_(n+m-i)~([m])(x)=∑_(i=0)~Nb_(i, n)Q_(n-i)(x), n≥0, where {M; N, and m are non-negative integers, {a_(i,n)}_(n≥0), 0≤i ≤ M, and {b_(i,n)}_(n≥0), 0 ≤ i ≤ N, are sequences of complex numbers such that a_(M,n) ≠ 0 if n ≥ M; b_(N,n) ≠ 0 if n ≥ N, and a_(i,n)= b_(i, n) = 0 if i > n. When m = 1, (U, V) is called an (M, N)-coherent pair. In this work, we give a matrix interpretation of (M, N)-coherent pairs of linear functionals. Indeed, an algebraic relation between the corresponding monic tridiagonal (Jacobi) matrices associated with such linear functionals is stated. As a particular situation, we analyze the case when one of the linear functionals is classical. Finally, the relation between the Jacobi matrices associated with (M, N)-coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product defined by the pair (U, V) is deduced.