Derangement polynomials with a complex variable

Abdelkader Benyattou
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310-5132, Online ISSN 2367-8275
Volume 26, 2020, Number 4, Pages 128—135
DOI: 10.7546/nntdm.2020.26.4.128-135
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Authors and affiliations

Abdelkader Benyattou
Department of Mathematics and Informatics, University of Djelfa, Algeria
RECITS Laboratory, P. O. 32 Box 32, El Alia 16111, Algiers, Algeria


In this paper, we define new polynomials with a complex variable related to the derangement polynomials and we give some properties of those polynomials. We use umbral calculus to establish a new congruence concerning the derangement polynomials with a complex variable.


  • Derangement polynomials
  • Complex variable, Congruence
  • Umbral calculus

2010 Mathematics Subject Classification

  • 11B83
  • 11A07
  • 30C10


  1. Benyattou, A., & Mihoubi, M. (2018). Curious congruences related to the Bell polynomials, Quaest. Math., 41(3), 437–448.
  2. Benyattou, A., & Mihoubi, M. (2019). Real-rooted polynomials via generalized Bell umbra. Notes on Number Theory and Discrete Mathematics, 25(2), 136–144.
  3. Darus, M., & Ibrahim, R. (2010). On generalisation of polynomials in complex plane, Advances in Decision Sciences, 2010, (2010), 9 pages.
  4. Gertsch, A., & Robert, A. M. (1996). Some congruences concerning the Bell numbers, Bull. Belg. Math. Soc. Simon Stevin, 3, 467–475.
  5. Gessel, I. M. (2003). Applications of the classical umbral calculus, Algebra Universalis, 49, 397–434.
  6. Kim, D. S., Kim, T., & Lee, H. (2019). A note on Degenerate Euler and Bernoulli polynomials, Symmetry, 11, 1168.
  7. Kim, T., & Kim, D. S. (2018). Some identities on derangement and degenerate derangement polynomials, Advances in Mathematical Inequalities and Applications, 265–277, Trends Math., Birkhauser/Springer, Singapore.
  8. Kim, T., Kim, D. S., Dolgy, D. V., & Kwon, J. (2018). Some identities of derangement numbers. Proc. Jangjeon Math. Soc., 21(1), 125–141.
  9. Kim, T., Kim, D. S., Kwon, H.-I., & Jang, L.-C. (2018). Fourier series of sums of products of r-derangement functions, J. Nonlinear Sci. Appl., 11(4), 575–590.
  10. Roman, S. (1984). The Umbral Calculus, Academic Press, Orlando, FL.
  11. Rota, G. C., & Taylor, B. D. (1994). The classical umbral calculus, SIAM J. Math. Anal., 25, 694–711.
  12. Sun, Z.-W., & Zagier, D. (2011). On a curious property of Bell numbers, Bull. Aust. Math. Soc., 84, 153–158.

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Cite this paper

Benyattou, A. (2020). Derangement polynomials with a complex variable. Notes on Number Theory and Discrete Mathematics, 26 (4), 128-135, doi: 10.7546/nntdm.2020.26.4.128-135.

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