Skip to main content
SpringerLink
Log in

Some identities of extended degenerate r-central Bell polynomials arising from umbral calculus

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Extended degenerate r-central factorial numbers of the second kind and extended degenerate r-central Bell polynomials were introduced recently, as a degenerate version, a central analogue and an r-extension of Stirling numbers of the second kind and Bell polynomials, respectively, and various properties of them were investigated. The purpose of the present paper is to further derive some properties, recurrence relations and identities concerning those numbers and polynomials. In particular, we will represent the extended degenerate r-central Bell polynomials in terms of a good number of well-known special polynomials.

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. Araci, S.: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 233, 599–607 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49(3), 241–259 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 15, 51–88 (1979)

    MathSciNet  MATH  Google Scholar 

  4. Carlitz, L.: Some remarks on the Bell numbers. Fibonnaci Quart. 18(1), 66–73 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Carlitz, L., Riodan, J.: The divided central differences of zero. Can. J. Math. 15, 94–100 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, Y., Ping, J.: Some recursion formulae for the number of derangements and Bell numbers. J. Math. Res. Appl. 36(1), 15–22 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Jang, G.-W., Kim, T.: A note on type 2 degenerate Euler and Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 29(1), 147–159 (2019)

    MATH  Google Scholar 

  8. Jang, L.-C., Kim, T., Kim, D.S., Kim, H.Y.: Extended \(r\)-central Bell polynomials with umbral calculus viewpoint. Adv. Differ. Equ. 2019, 202 (2019)

    Article  MATH  Google Scholar 

  9. Kim, D.S., Dolgy, D.V., Kim, D., Kim, T.: Some identities on \(r\)-central factorial numbers and \(r\)-central Bell polynomials. Adv. Differ. Equ. 2019, 245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kim, D.S., Dolgy, D.V., Kim, T., Kim, D.: Extended degenerate \(r\)-central factorial numbers of the second kind and extended degenerate \(r\)-central Bell polynomials. Symmetry 11(4), 595 (2019)

    Article  MATH  Google Scholar 

  11. Kim, D.S., Jang, G.-W., Kwon, H.-I., Kim, T.: Two variable higher-order degenerate Fubini polynomials. Proc. Jangjeon Math. Soc. 21(1), 5–22 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Kim, D.S., Kim, T.: Identities involving \(r\)-Stirling numbers. J. Comput. Anal. Appl. 17(4), 674–680 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Kim, D.S., Kim, T.: Some identities of Bell polynomials. Sci. China Math. 58(10), 2095–2104 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, T.: A note on central factorial numbers¿ Proc. Jangjeon Math. Soc. 21(4), 575–588 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Kim, T., Kim, D.S.: Degenerate central Bell numbers and polynomials. Rev. R. Acad. de Cienc. Exactas, Fís. Nat. Ser. A Math. RACSAM. 113, 2507–2513 (2019). https://doi.org/10.1007/s13398-019-00637-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Kim, T., Kim, D.S.: Degenerate central factorial numbers of the second kind. Rev. R. Acad. de Cienc. Exactas, Fís. Nat. Ser. A Math. RACSAM. 113, 3359–3367 (2019). https://doi.org/10.1007/s13398-019-00700-w

    Article  MathSciNet  MATH  Google Scholar 

  17. Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Extended central facotrial polynomials of the second kind. Adv. Differ. Equ. 2019, 24 (2019)

    Article  MATH  Google Scholar 

  18. Kim, T., Kim, D.S., Kwon, H.-I., Kwon, J.: Umbral calculus approach to \(r\)-Stirling numbers of the second kind and \(r\)-Bell polynomials. J. Comput. Anal. Appl. 27(1), 173–188 (2019)

    Google Scholar 

  19. Kim, T., Kim, D.S., Kwon, J., Dolgy, D.V.: A note on central Bell numbers and polynomials. to appear in Russ. J. Math. Phys

  20. Kim, T., Yao, Y., Kim, D.S., Jang, G.-W.: Degenerate \(r\)-Stirling numbers and \(r\)-Bell polynomials. Russ. J. Math. Phys. 25(1), 44–58 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krzywonos, N., Alayont, F.: Rook polynomials in higher dimensions. Student Summer Scholars 29, (2009)

  22. Quaintance, J., Gould, H.W.: Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H.W. Gould, World Scientific Publishing Co. Pte. Ltd., Singapore (2016)

  23. Riordan, J.: Combinatorial identities. Wiley, New York-London-Sydney (1968)

    MATH  Google Scholar 

  24. Roman, S.: The Umbral Calculus. Academic Press, New York (1984)

    MATH  Google Scholar 

  25. Simsek, Y.: Identities and relations related to combinatorial numbers and polynomials. Proc. Jangjeon Math. Soc. 20(2), 127–135 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, W.: Some identities involving the Euler and the central factorial numbers. Fibonacci Quart. 36(2), 154–157 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Xian Technological University, Xian, 710021, China

    Taekyun Kim

  2. Department of Mathematics, Kwangwoon University, Seoul, 01897, Republic of Korea

    Taekyun Kim

  3. Department of Mathematics, Sogang University, Seoul, 04107, Republic of Korea

    Dae San Kim

Authors
  1. Taekyun Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dae San Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taekyun Kim.

Additional information

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

Kim, T., Kim, D.S. Some identities of extended degenerate r-central Bell polynomials arising from umbral calculus. RACSAM 114, 1 (2020). https://doi.org/10.1007/s13398-019-00732-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-019-00732-2

Keywords

Mathematics Subject Classification

Search

Navigation