Shifted Darboux Transformations of the Generalized Jacobi Matrices, I

Let ℑ 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 ℑ to admit a factorization ℑ = 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 ℑ defined by ℑ(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 ℑ(p) are found. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.