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Several formulas for Bernoulli numbers and polynomials

  • * Corresponding author: Bijan Kumar Patel

    * Corresponding author: Bijan Kumar Patel 
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  • A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in the expansion $ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $, where $ a \neq 0, b $ are complex numbers, have studied in [16]. In this paper, we show that Bernoulli polynomials $ B_{p}(x) $ can be written in terms of the numbers $ S_{1,x}\left(p,k\right) $, and then use the known results for $ S_{1,x}\left(p,k\right) $ to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.

    Mathematics Subject Classification: Primary: 11B68, 11B73; Secondary: 11B83.


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  • [1] L. Carlitz, Problem 795, Math. Mag., 44 (1971), 107.
    [2] L. Comtet, Advanced Combinatorics, Reidel, 1974.
    [3] H. W. Gould, Tables of Combinatorial Identities, Edited and compiled by Prof. Jocelyn Quaintance, 2010. Available from: https://math.wvu.edu/~hgould/.
    [4] H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79 (1972), 44-51.  doi: 10.1080/00029890.1972.11992980.
    [5] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, 2$^nd$ edition, Addison-Wesley, 1994.
    [6] B. N. GuoI. Mezö and F. Qi, An explicit formula for Bernoulli polynomials in terms of $r$-Stirling numbers of the second kind, Rocky Mountain J. Math., 46 (2016), 1919-1923.  doi: 10.1216/RMJ-2016-46-6-1919.
    [7] B. N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Ana. Num. Theor., 3 (2015), 27-30. 
    [8] B. C. Kellner, Identities between polynomials related to Stirling and harmonic numbers, Integers, 14 (2014), 54-76. 
    [9] B. Mazur, Bernoulli Numbers and the Unity of Mathematics, Available from: http://people.math.harvard.edu/ mazur/papers/slides.Bartlett.pdf
    [10] M. Merca, A new connection between $r$-Whitney numbers and Bernoulli polynomials, Integral Transforms Spec. Funct., 25 (2014), 937-942.  doi: 10.1080/10652469.2014.940580.
    [11] M. Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J. Number Theory, 151 (2015), 223-229.  doi: 10.1016/j.jnt.2014.12.024.
    [12] M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period. Math. Hungar., 73 (2016), 259-264.  doi: 10.1007/s10998-016-0140-5.
    [13] M. Merca, On lacunary recurrences with gaps of length four and eight for the Bernoulli numbers, Bull. Korean Math. Soc., 56 (2019), 491-499.  doi: 10.4134/BKMS.b180347.
    [14] M. Merca, Bernoulli numbers and symmetric functions, Rev. R. Acad. Cienc. Exactas F$\acute{{i}}$s. Nat. (Esp.), Serie A, Matem$\acute{{a}}$ticas, 114 (2020), 20–36. doi: 10.1007/s13398-019-00774-6.
    [15] I. Mező, A new formula for the Bernoulli polynomials, Results Math., 58 (2010), 329-335.  doi: 10.1007/s00025-010-0039-z.
    [16] C. Pita-Ruiz, Generalized stirling Numbers I, preprint, arXiv: 1803.05953v1.
    [17] C. Pita-Ruiz, Carlitz-Type and other Bernoulli Identities, J. Integer Seq., 19 (2016), 27 pp.
    [18] F. A. Shiha, An explicit formula for Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers, J. Ana. Num. Theor., 6 (2018), 47-50. 
    [19] J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, J. Reine Angew. Math., 94 (1883), 203-232.  doi: 10.1515/crll.1883.94.203.
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