An identity involving Bernoulli numbers and the Stirling numbers of the second kind

Sumit Kumar Jha
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 160–162
DOI: 10.7546/nntdm.2020.26.3.160-162
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Sumit Kumar Jha
International Institute of Information Technology
Hyderabad-500 032, India

Abstract

Let Bn denote the Bernoulli numbers, and S(n, k) denote the Stirling numbers of the second kind. We prove the following identity
B_{m+n}=\sum_{\substack{0\leq k \leq n \\ 0\leq l \leq m}}\frac{(-1)^{k+l}\,k!\, l!\, S(n,k)\,S(m,l)}{(k+l+1)\,\binom{k+l}{l}}.
To the best of our knowledge, the identity is new.

Keywords

  • Bernoulli numbers
  • Stirling numbers of the second kind
  • Riemann zeta function
  • Polylogarithm function

2010 Mathematics Subject Classification

  • 11B68
  • 11B73

References

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  2. Wolfram Research (2020). Polylogarithm: Integration (formula 10.08.21.0027.01). Available at: https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/21/02/03/0005/.
  3. Gould, H. W. (1972). Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79 (1), 44–51.
  4. Landsburg, S. E. (2009). Stirling numbers and polylogarithms, preprint. Available at: http://www.landsburg.com/query.pdf.
  5. Qi, F., & Guo, B. N. (2014). Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin), 34 (3), 311–317.

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Cite this paper

Jha, S. K. (2020). An identity involving Bernoulli numbers and the Stirling numbers of the second kind. Notes on Number Theory and Discrete Mathematics, 26 (3), 160-162, DOI: 10.7546/nntdm.2020.26.3.160-162.

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