Skip to main content
SpringerLink
Log in

Several Continued Fraction Expansions of Generalized Cauchy Numbers

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

We consider two types of continued fraction expansions of the generating functions of Cauchy numbers. One type has been often studied by many authors, but another has not. In this paper, we give several continued fraction expansions of hypergeometric Cauchy numbers, shifted Cauchy numbers and leaping Cauchy numbers. In special cases, continued fraction expansions of the classical Cauchy numbers of both kinds are deduced.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974)

    Book  Google Scholar 

  2. Dey, P. K., Komatsu, T.: Some identities of Cauchy numbers associated with continued fractions. Res. Math. 74 (2019), Article 83, 11 pp

  3. Flajolet, P.: Combinatorial aspects of continued fractions. Discrete Math. 32, 125–161 (1980)

    Article  MathSciNet  Google Scholar 

  4. Flajolet, P.: Combinatorial aspects of continued fractions (Reprint). Discrete Math. 306(10–11), 992–1021 (2006)

    Article  Google Scholar 

  5. Komatsu, T.: Poly-Cauchy numbers. Kyushu J. Math. 67, 143–153 (2013)

    Article  MathSciNet  Google Scholar 

  6. Komatsu, T.: Hypergeometric Cauchy numbers. Int. J. Number Theory 9, 545–560 (2013)

    Article  MathSciNet  Google Scholar 

  7. Komatsu, T.: Incomplete poly-Cauchy numbers. Monatsh. Math. 180, 271–288 (2016)

    Article  MathSciNet  Google Scholar 

  8. Komatsu, T.: Shifted Cauchy numbers. Filomat 32(18), 6167–6176 (2018)

    Article  MathSciNet  Google Scholar 

  9. Komatsu, T.: Continued fraction expansions of the generating functions of Bernoulli and related numbers. Indag. Math. 31, 695–713 (2020)

    Article  MathSciNet  Google Scholar 

  10. Komatsu, T., Pita-Ruiz, C.: Leaping Cauchy numbers. Quaest. Math. 43, 213–226 (2020)

    Article  MathSciNet  Google Scholar 

  11. Loya, P.: Amazing and Aesthetic Aspects of Analysis (Undergraduate Texts in Mathematics). Springer, Berlin (2017)

    Book  Google Scholar 

  12. Merlini, D., Sprugnoli, R., Verri, M.C.: The Cauchy numbers. Discrete Math. 306, 1906–1920 (2006)

    Article  MathSciNet  Google Scholar 

  13. Sloane, N. J. A.: The on-line encyclopedia of integer sequences, available at oeis.org. (2021)

  14. Polya, G.: Induction and Analogy in Mathematics, Mathematics and Plausible Reasoning, vol. I. Princeton University Press, Princeton (1954)

    MATH  Google Scholar 

  15. Wall, H.S.: Analytic Theory of Continued Fractions. Chelsea, New-York (1967)

    MATH  Google Scholar 

  16. Zhu, B.-X.: Positivity of iterated sequences of polynomials. SIAM J. Discrete Math. 32(3), 1993–2010 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referee for the comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Takao Komatsu

Authors
  1. Takao Komatsu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takao Komatsu.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Komatsu, T. Several Continued Fraction Expansions of Generalized Cauchy Numbers. Bull. Malays. Math. Sci. Soc. 44, 2425–2446 (2021). https://doi.org/10.1007/s40840-021-01074-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01074-2

Keywords

Mathematics Subject Classification

Search

Navigation