I. Kovalyov

A sequence of the real numbers $\textbf{s}=\{s_{i}\}_{i=0}^{\ell}$ is associated with the some indefinite Stieltjes moment problem and generalized Jacobi matrices. The relation between the $\alpha-$regular indefinite Stieltjes moment problem and shifted Darboux transformation of the generalized Jacobi matrix is studied. The new formulas for the Stieltjes polynomials with the shift are found and one are used to obtain the description of the solutions of the $\alpha-$regular indefinite Stieltjes moment problem.

Послідовність дійсних чисел $\textbf{s}=\{s_{i}\}_{i=0}^{\ell}$ пов'язана з деякою задачею про невизначений момент Стілтьєса та узагальненими матрицями Якобі. Досліджено зв'язок між $\alpha-$регулярною проблемою невизначеного моменту Стілтьєса та зміщеним перетворенням Дарбу узагальненої матриці Якобі. Знайдено нові формули для поліномів Стілтьєса зі зсувом та використано для отримання опису розв’язків $\alpha-$регулярної невизначеної проблеми моменту Стілтьєса.

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulas for diagonal and sub-diagonal Padé approximants for formal power series corresponding to indefinite Stieltjes moment problem and convergence results for Padé approximants are presented.

With each sequence of real numbers ${\mathbf s}=\{s_j\}_{j=0}^\infty$ two kinds of continued fractions are associated, - the so-called $P-$fraction and a generalized Stieltjes fraction that, in the case when ${\mathbf s}=\{s_j\}_{j=0}^\infty$ is a sequence of moments of a probability measure on $\mathbb R_+$, coincide with the $J-$fraction and the Stieltjes fraction, respectively. A subclass $\mathcal H^{reg}$ of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For ${\mathbf s}\in\mathcal H^{reg}$ the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.

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.