I. Kovalyov

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Articles: 2

On a class of generalized Stieltjes continued fractions

Vladimir Derkach, Ivan Kovalyov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 21 (2015), no. 4, 315-335

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.

Darboux transformation of generalized Jacobi matrices

Ivan Kovalyov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 20 (2014), no. 4, 301-320

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.

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