Skip to main content
Log in

Explicit formulae for sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We study sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials. As a main result, for any positive integer \(m\), explicit expressions of sums of \(m\) products are given. This result extends that of the first author, as well as the famous Euler formula about sums of two products of Bernoulli numbers.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bayad, A., Hamahata, Y.: Polylogarithms and poly-Bernoulli polynomials. Kyushu J. Math. 65, 15–24 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Comtet, L.: Advanced Combinatorics. Reidel, Doredecht (1974)

    Book  MATH  Google Scholar 

  3. Coppo, M.-A., Candelpergher, B.: The Arakawa–Kaneko zeta functions. Ramanujan J. 22, 153–162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dilcher, K.: Sums of products of Bernoulli numbers. J. Number Theory 60, 23–41 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  6. Hansen, E.R.: A Table of Series and Products. Prentice Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  7. Sharp, H. Jr.: Cardinality of finite topologies. J. Comb. Theory 5, 82–86 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kamano, K.: Sums of products of Bernoulli numbers, including poly-Bernoulli numbers. J. Integer Seq. 13 (2010). Article 10.5.2

  9. Kaneko, M.: Poly-Bernoulli numbers. J. Théor. Nr. Bordx. 9, 199–206 (1997)

    Article  Google Scholar 

  10. Nölund, N.E.: Mémoire sur les polynomes de Bernoulli. Acta Math. 43, 121–196 (1922)

    Article  MathSciNet  Google Scholar 

  11. Sasaki, Y.: On generalized poly-Bernoulli numbers and related \(L\)-functions. J. Number Theory 132, 156–170 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ken Kamano.

Additional information

The second author was supported in part by the Grant-in-Aid for Scientific research (C) (No. 22540005), the Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamano, K., Komatsu, T. Explicit formulae for sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials. Ramanujan J 33, 301–313 (2014). https://doi.org/10.1007/s11139-013-9509-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9509-8

Keywords

Mathematics Subject Classification (2010)

Navigation