A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers

  • Authors

    • Bai-Ni Guo Henan Polytechnic University
    • Feng Qi Department of Mathematics, College of Science, Tianjin Polytechnic UniversityTianjin City, 300160, China
    2015-02-14
    https://doi.org/10.14419/gjma.v3i1.4168
  • Explicit Formula, Bernoulli Number, Genocchi Number, Stirling Number of The Second Kind.

  • Abstract

    In the paper, the authors concisely review some explicit formulas and establish a new explicit formula for the Bernoulli and Genocchi numbers in terms of theStirlingnumbers of the second kind.

    Author Biography

    • Feng Qi, Department of Mathematics, College of Science, Tianjin Polytechnic UniversityTianjin City, 300160, China

  • References

    1. [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co.,Dordrecht andBoston, 1974.

      [2] H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79 (1972), 44--51.

      [3] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics---A Foundation for Computer Science, Addison-Wesley Publishing Company, Advanced Book Program,Reading,MA, 1989.

      [4] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics---A Foundation for Computer Science, nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.

      [5] B.-N. Guo and F. Qi, A new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers, available online at http://arxiv.org/abs/1407.7726.

      [6] B.-N. Guo, I. Mező, and F. Qi, An explicit formula for Bernoulli polynomials in terms of -Stirling numbers of the second kind, available online at http://arxiv.org/abs/1402.2340.

      [7] B.-N. Guo and F. Qi, Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin) 34 (2014), no. 2, 187--193; Available online at http://dx.doi.org/10.1515/anly-2012-1238.

      [8] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory 3 (2015), no. 1, 27--30; Available online at http://dx.doi.org/10.12785/jant/030105.

      [9] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math. 272 (2014), 251--257; Available online at http://dx.doi.org/10.1016/j.cam.2014.05.018.

      [10] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math. 255 (2014), 568--579; Available online at http://dx.doi.org/10.1016/j.cam.2013.06.020.

      [11] S.-L. Guo and F. Qi, Recursion formulae for , Z. Anal. Anwendungen 18 (1999), no. 4, 1123--1130; Available online at http://dx.doi.org/10.4171/ZAA/933.

      [12] S. Jeong, M.-S. Kim, and J.-W. Son, On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic, J. Number Theory 113 (2005), no. 1, 53--68; Available online at http://dx.doi.org/10.1016/j.jnt.2004.08.013.

      [13] J. Higgins, Double series for the Bernoulli and Euler numbers, J. London Math. Soc. nd Ser. 2 (1970), 722--726; Available online at http://dx.doi.org/10.1112/jlms/2.Part_4.722.

      [14] B. F. Logan, Polynomials related to the Stirling numbers, AT&T Bell Laboratories internal technical memorandum, August 10, 1987.

      [15] F. Qi, Derivatives of tangent function and tangent numbers, available online at http://arxiv.org/abs/1202.1205.

      [16] F. Qi and B.-N. Guo, Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin) 34 (2014), no. 3, 311--317; Available online at http://dx.doi.org/10.1515/anly-2014-0003.

      [17] S. Shiraiand K.-I. Sato, Some identities involving Bernoulli and Stirling numbers, J. Number Theory 90 (2001), no. 1, 130--142; Available online at http://dx.doi.org/10.1006/jnth.2001.2659.

      [18] A.-M. Xu and Z.-D. Cen, Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math. 260 (2014), 201--207; Available online at http://dx.doi.org/10.1016/j.cam.2013.09.077.

  • Downloads

    Additional Files

  • How to Cite

    Guo, B.-N., & Qi, F. (2015). A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers. Global Journal of Mathematical Analysis, 3(1), 33-36. https://doi.org/10.14419/gjma.v3i1.4168

    Received date: 2015-01-13

    Accepted date: 2015-02-10

    Published date: 2015-02-14