# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021006
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## Several formulas for Bernoulli numbers and polynomials

 1 Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China 2 Department of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar 751024, India 3 Universidad Panamericana. Facultad de Ingeniería., Augusto Rodin 498, Ciudad de México, 03920, México

* Corresponding author: Bijan Kumar Patel

Received  September 2020 Revised  January 2021 Early access March 2021

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.

Citation: Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, doi: 10.3934/amc.2021006
##### References:
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##### References:
 [1] L. Carlitz, Problem 795, Math. Mag., 44 (1971), 107. Google Scholar [2] L. Comtet, Advanced Combinatorics, Reidel, 1974.  Google Scholar [3] H. W. Gould, Tables of Combinatorial Identities, Edited and compiled by Prof. Jocelyn Quaintance, 2010. Available from: https://math.wvu.edu/~hgould/. Google Scholar [4] H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79 (1972), 44-51.  doi: 10.1080/00029890.1972.11992980.  Google Scholar [5] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, 2$^nd$ edition, Addison-Wesley, 1994.  Google Scholar [6] B. N. Guo, I. 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.  Google Scholar [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.   Google Scholar [8] B. C. Kellner, Identities between polynomials related to Stirling and harmonic numbers, Integers, 14 (2014), 54-76.   Google Scholar [9] B. Mazur, Bernoulli Numbers and the Unity of Mathematics, Available from: http://people.math.harvard.edu/ mazur/papers/slides.Bartlett.pdf Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] I. Mező, A new formula for the Bernoulli polynomials, Results Math., 58 (2010), 329-335.  doi: 10.1007/s00025-010-0039-z.  Google Scholar [16] C. Pita-Ruiz, Generalized stirling Numbers I, preprint, arXiv: 1803.05953v1. Google Scholar [17] C. Pita-Ruiz, Carlitz-Type and other Bernoulli Identities, J. Integer Seq., 19 (2016), 27 pp.  Google Scholar [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.   Google Scholar [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.  Google Scholar
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