Darboux transformation with parameter of generalized Jacobi matrices.

A monic generalized Jacobi matrix $$ \mathfrak{J} $$ is factorized into upper and lower triangular two-diagonal block matrices of special forms so that J = UL. It is shown that such factorization depends on a free real parameter d(∈ ℝ). As the main result, it is shown that the matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)}= LU $$ is also a monic generalized Jacobi matrix. The matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ is called the Darboux transform of $$ \mathfrak{J} $$ with parameter d. An analog of the Geronimus formula for polynomials of the first kind of the matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ is proved, and the relations between m-functions of J and $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ are found. [ABSTRACT FROM AUTHOR]