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Zh. Vychisl. Mat. Mat. Fiz., 2018, Volume 58, Number 11, Pages 1747–1770 (Mi zvmmf10843)  

This article is cited in 1 scientific paper (total in 1 paper)

Factorial transformation for some classical combinatorial sequences

V. P. Varin

Keldysh Institute of Applied Mathematics RAS, Moscow, Russia

Abstract: Factorial transformation known from Euler's time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Padé approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.

Key words: factorial transformation, factorial series, continued fractions, Stirling, Bernoulli, Bell, Euler and Tangent numbers, divergent power series, generating functions.

DOI: https://doi.org/10.31857/S004446690003530-4

References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2018, 58:11, 1687–1707

Bibliographic databases:

UDC: 519.624.2
Received: 02.04.2018

Citation: V. P. Varin, “Factorial transformation for some classical combinatorial sequences”, Zh. Vychisl. Mat. Mat. Fiz., 58:11 (2018), 1747–1770; Comput. Math. Math. Phys., 58:11 (2018), 1687–1707

Citation in format AMSBIB
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\pages 1687--1707
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. P. Varin, “Invariantnye krivye nekotorykh diskretnykh dinamicheskikh sistem”, Preprinty IPM im. M. V. Keldysha, 2020, 078, 27 pp.  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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