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:
[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.

show all references

References:
[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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