VISKOVATOV-TYPE ALGORITHM FOR EXPANDING FORMAL TRIPLE POWER SERIES INTO THREE-DIMENSIONAL CORRESPONDING REGULAR CONTINUED FRACTION
DOI:
https://doi.org/10.15407/dopovidi2025.05.062Keywords:
Viskovatov’s algorithm, triple power series, three-dimensional corresponding regular continued fractionAbstract
In this paper, we propose an algorithm for decomposing a formal triple power series into a three-dimensional regular continuous fraction corresponding to this series, which generalises one of the simplest algorithms for decomposing a formal power series into a corresponding continuous fraction, namely, Viskovatov’s algorithm. The concept of correspondence between formal power series and continued fractions leads to the construction of representations of analytic functions using continued fractions. The construction of algorithms for calculating continued fractions from the known coefficients of the corresponding formal power series is one of the main tasks of the analytical theory of continued fractions and their multidimensional generalisations. The most systematic study of fractions corresponding to formal power series and the order of their correspondence was obtained by W. Jones and W. Thron. The algorithms for the expanding into corresponding fractions with different orders of correspondence used Hankel’s determinants for computing the coefficients of such fractions. For multidimensional generalizations of continuous fractions, namely for the construction of algorithms for the expansion a formal multiple power series into multidimensional continued fractions, the Hankel-type determinants are also used, but their computation with an increase of the number of variables is essentially complicated. The Viskovatov-type algorithm is proposed, which, from a computational point of view, provides an effective solution to the problem of decomposing a formal triple power series into a three-dimensional regular continuous fraction corresponding to this series.
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