Open Access

# On a class of generalized Stieltjes continued fractions

### Abstract

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.

Key words: Moment problem, continued fraction, generalized Stieltjes fraction, Darboux transformation, generalized Jacobi matrix, triangular factorization, unwrapping transformation.

### Article Information

 Title On a class of generalized Stieltjes continued fractions Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 315-335 MathSciNet MR3469531 zbMATH 06630277 Milestones Received 02/02/2015; Revised 05/03/2015 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

Department of Mathematics, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka,Ivano-Frankivsk, 76018, Ukraine

Ivan Kovalyov
Department of Mathematics, Dragomanov National Pedagogical University, 9 Pirogova, Kyiv, 01601, Ukraine

### Citation Example

Vladimir Derkach and Ivan Kovalyov, On a class of generalized Stieltjes continued fractions, Methods Funct. Anal. Topology 21 (2015), no. 4, 315-335.

### BibTex

@article {MFAT781,
AUTHOR = {Derkach, Vladimir and Kovalyov, Ivan},
TITLE = {On a class of generalized Stieltjes continued fractions},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {4},
PAGES = {315-335},
ISSN = {1029-3531},
MRNUMBER = {MR3469531},
ZBLNUMBER = {06630277},
URL = {http://mfat.imath.kiev.ua/article/?id=781},
}

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