Darboux transformation of generalized Jacobi matrices

Abstract

Let $\mathfrak{J}$ be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of special form, introduced by M.~Derevyagin and V.~Derkach in 2004. We find conditions for a monic generalized Jacobi matrix $\mathfrak{J}$ to admit a factorization $\mathfrak{J}=\mathfrak{LU}$ with $\mathfrak{L}$ and $\mathfrak{U}$ being lower and upper triangular two-diagonal block matrices of special form. In this case the Darboux transformation of $\mathfrak{J}$ defined by $\mathfrak{J}^{(p)}=\mathfrak{UL}$ is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation $\mathfrak{J}^{(p)}$ are found.