Skip to main content
SpringerLink
Log in

Explicit Formulas for Special Values of the Bell Polynomials of the Second Kind and for the Euler Numbers and Polynomials

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In the paper, the authors establish by two approaches several explicit formulas for special values of the Bell polynomials of the second kind, derive explicit formulas for the Euler numbers and polynomials in terms of double sums and the weighted Stirling numbers, and find a property for special values of the Bell polynomials of the second kind.

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: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition. D. Reidel Publishing Co., Dordrecht (1974)

    Book  MATH  Google Scholar 

  2. Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49(3), 241–259 (1984). doi:10.1016/0012-365X(84)90161-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlitz, L.: Weighted Stirling numbers of the first and second kind. I. Fibonacci Quart. 18(2), 147–162 (1980)

    MathSciNet  MATH  Google Scholar 

  4. Carlitz, L.: Weighted Stirling numbers of the first and second kind. II. Fibonacci Quart. 18(3), 242–257 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Guo, B.-N., Mező, I., Qi, F.: An explicit formula for the Bernoulli polynomials in terms of the \(r\)-Stirling numbers of the second kind. Rocky Mountain J. Math. 46(6), 1919–1923 (2016). doi:10.1216/RMJ-2016-46-6-1919

    Article  MathSciNet  MATH  Google Scholar 

  6. Nörlund, N.: Vorlesungen über Differenzenrechnung. Chelsea, New York (1954)

    Google Scholar 

  7. Howard, F.T.: Congruences and recurrences for Bernoulli numbers of higher order. Fibonacci Quart. 32(4), 316–328 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Yakubovich, S.: Certain identities, connection and explicit formulas for the Bernoulli, Euler numbers and Riemann zeta-values. arXiv preprint (2014). arXiv:1406.5345

  9. Guo, B.-N., Qi, F.: Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind. J. Comput. Appl. Math. 272, 251–257 (2014). doi:10.1016/j.cam.2014.05.018

    Article  MathSciNet  MATH  Google Scholar 

  10. Qi, F.: Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 268, 844–858 (2015). doi:10.1016/j.amc.2015.06.123

    MathSciNet  Google Scholar 

  11. Carlitz, L.: Eulerian numbers and polynomials. Math. Mag. 32, 247–260 (1958/1959)

  12. Qi, F., Zheng, M.-M.: Explicit expressions for a family of the Bell polynomials and applications. Appl. Math. Comput. 258, 597–607 (2015). doi:10.1016/j.amc.2015.02.027

    MathSciNet  MATH  Google Scholar 

  13. Qi, F.: Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind. Filomat 28(2), 319–327 (2014). doi:10.2298/FIL1402319O

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, B.-N., Qi, F.: An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind. J. Anal. Number Theory 3(1), 27–30 (2015). doi:10.12785/jant/030105

    Google Scholar 

  15. Qi, F.: Diagonal recurrence relations for the Stirling numbers of the first kind. Contrib. Discrete Math. 11(1), 22–30 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Qi, F., Shi, X.-T., Liu, F.-F., Kruchinin, D.V.: Several formulas for special values of the Bell polynomials of the second kind and applications. J. Appl. Anal. Comput. 7(3), 857–871 (2017). doi:10.11948/2017054

  17. Howard, F.T.: A special class of Bell polynomials. Math. Comput. 35(151), 977–989 (1980). doi:10.2307/2006208

    Article  MathSciNet  MATH  Google Scholar 

  18. Qi, F., Guo, B.-N.: Viewing some ordinary differential equations from the angle of derivative polynomials. Preprints 2016, 2016100043, 12 pages. doi:10.20944/preprints201610.0043.v1

  19. Wei, C.-F., Qi, F.: Several closed expressions for the Euler numbers. J. Inequal. Appl. 2015, 219 (2015). doi:10.1186/s13660-015-0738-9. (8 pages)

    Article  MathSciNet  MATH  Google Scholar 

  20. Higgins, J.: Double series for the Bernoulli and Euler numbers. J. Lond. Math. Soc. (2) 2, 722–726 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, B.-N., Qi, F.: Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers. ResearchGate Technical Report (2015). doi:10.13140/2.1.3794.8808

Download references

Acknowledgements

The authors are grateful to several expert mathematicians and anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Feng Qi

  2. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, Inner Mongolia Autonomous Region, China

    Feng Qi

  3. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Feng Qi

  4. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Bai-Ni Guo

Authors
  1. Feng Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bai-Ni Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Qi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, F., Guo, BN. Explicit Formulas for Special Values of the Bell Polynomials of the Second Kind and for the Euler Numbers and Polynomials. Mediterr. J. Math. 14, 140 (2017). https://doi.org/10.1007/s00009-017-0939-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-0939-1

Mathematics Subject Classification

Keywords

Search

Navigation