Darboux transformation of generalized Jacobi matrices. (English) Zbl 1324.47058

The author studies the so-called monic generalized Jacobi matrices, infinite block matrices \(J\) associated with systems of difference equations of the form \[ b_j y_{n_j-1}(\lambda)-p_j(\lambda)y_{n_j}(\lambda)+y_{n_j+1}(\lambda)=0, \] where \(\{n_j\}\) is a sequence connected with singularity levels of submatrices of the Hankel matrix defined by a sequence of moments, \(b_j\in \mathbb R\setminus \{ 0\}\), \(p_j(\lambda)\) are monic polynomials of degree \(l_j=n_{j+1}-n_j\).
Such a study was carried out by M. Derevyagin and V. Derkach [Linear Algebra Appl. 435, No. 12, 3056–3084 (2011; Zbl 1227.47018)] under an additional quasi-definiteness condition. In the paper under review, factorizations of \(J\) into the product of lower and upper triangular matrices are found, as well as analogs of the Darboux transformation and of the Christoffel formulas for polynomials of the first and second kind.


47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47A57 Linear operator methods in interpolation, moment and extension problems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
44A60 Moment problems


Zbl 1227.47018