Abstract
The definition, in previous studies, of vector Stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. The subject was more undefined in the matrix case. A new version of the QD algorithm for matrix problem, allows to extend to the vector and matrix cases the result of Stieltjes, expansion of a (scalar) function in terms of a Stieltjes continued fraction. Beside this connection, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. Finally, as a consequence, we extend to some dynamical systems a method known for Toda lattices.
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