doi: 10.3934/amc.2021006

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 Published  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. 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. 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.  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

show all references

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. 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.  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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