Explicit formulas for Euler polynomials and Bernoulli numbers

Laala Khaldi, Farid Bencherif and Miloud Mihoubi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 80—89
DOI: 10.7546/nntdm.2021.27.4.80-89
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Authors and affiliations

Laala Khaldi
Department of Mathematics, University of Bouira,
10000 Bouira, Algeria
Laboratory EDPNL&HM, Department of Mathematics,
ENS, BP 92, Vieux-Kouba, Algeria

Farid Bencherif
Laboratory LA3C, Faculty of Mathematics, USTHB
BP 32, El Alia , 16111, Algiers, Algeria

Miloud Mihoubi
Laboratory RECITS, Faculty of Mathematics, USTHB
BP 32, El Alia, 16111, Algiers, Algeria


In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.


  • Appell polynomials
  • Euler polynomials
  • Bernoulli numbers
  • Binomial coefficients

2020 Mathematics Subject Classification

  • 11B68
  • 05A10


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Cite this paper

Khaldi, L., Bencherif, F., & Mihoubi, M. (2021). Explicit formulas for Euler polynomials and Bernoulli numbers. Notes on Number Theory and Discrete Mathematics, 27(4), 80-89, doi: 10.7546/nntdm.2021.27.4.80-89.

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