Shifted Darboux transformations of the generalized Jacobi matrices, I

Abstract

Let J be a monic generalized Jacobi matrix, i.e., a three-diagonal block matrix of a special form. We find conditions for a monic generalized Jacobi matrix J to admit a factorization J = LU + αI with L and U being lower and upper triangular two-diagonal block matrices of special forms. In this case, the shifted parameterless Darboux transformation of J defined by J(p) = UL+αI is shown to be also a monic generalized Jacobi matrix. Analogs of the Christoffel formulas for polynomials of the first and second kinds corresponding to the Darboux transformation J(p) are found.