Skip to main content
SpringerLink
Log in

On q-generalized (r, s)-Stirling numbers

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper by using q-numbers and r-shift, the (q, r)-Stirling numbers with level s are studied. One of the main aims is to give several identities in their transforms. We also give some applications to the values of a certain kind of q-multiple zeta functions.

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

Notes

  1. Tweedie [14] called them the generalized Stirling numbers. But as there are many generalized Stirling numbers nowadays. So, in [6, 7], they are named differently.

References

  1. Bradley, D.M.: Multiple \(q\)-zeta values. J. Algebra 283(2), 752–798 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, Y., Readdy, M.: \(q\)-Stirling numbers: a new view. Adv. Appl. Math. 86, 50–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ernst, T.: \(q\)-Stirling numbers, an umbral approach. Adv. Dyn. Syst. Appl. 3(2), 251–282 (2008)

    MathSciNet  Google Scholar 

  5. Komatsu, T.: On \(s\)-Stirling transform and poly-Cauchy numbers of the second kind with level \(2\). Aequationes Math. 97(1), 31–61 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  6. Komatsu, T., Ramirez, J.L., Villamizar, D.: A combinatorial approach to the Stirling numbers of the first kind with higher level. Stud. Sci. Math. Hung. 58, 293–307 (2021). https://doi.org/10.1556/012.2021.58.3.1500

    Article  MathSciNet  MATH  Google Scholar 

  7. Komatsu, T., Ramirez, J.L., Villamizar, D.: A combinatorial approach to the generalized central factorial numbers. Mediterr. J. Math. 18, 14 (2021). https://doi.org/10.1007/s00009-021-01830-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Merca, M.: A \(q\)-analogue for sums of powers. Acta Arith. 183(2), 185–189 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Merca, M.: Some notes on the \((q, t)\)-Stirling numbers. Discrete Math. 342, 628–634 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)-MZSV sum. J. Num. Theory 132(1), 144–155 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schlesinger, K.-G.: Some remarks on \(q\)-deformed multiple polylogarithms, arXiv:math/0111022, (2001)

  12. Sloane, N. J. A.: The on-line encyclopedia of integer sequences, available at oeis.org. (2023)

  13. Spivey, M.Z.: Combinatorial sums and finite differences. Discrete Math. 307, 3130–3146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tweedie, C.: The Stirling numbers and polynomials. Proc. Edinburgh Math. Soc. 37, 2–25 (1918)

    Article  Google Scholar 

  15. Zhao, J.: Multiple \(q\)-zeta functions and multiple \(q\)-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zudilin, W.: Algebraic relations for multiple zeta values (Russian). Uspekhi Mat. Nauk 58(1), 3–32 (2003)

    MathSciNet  Google Scholar 

Download references

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

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Komatsu, T. On q-generalized (r, s)-Stirling numbers. Aequat. Math. (2023). https://doi.org/10.1007/s00010-023-00997-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00010-023-00997-4

Keywords

Mathematics Subject Classification

Search

Navigation