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

**Download full paper: PDF, 156 Kb**

## Details

### 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
*

### Abstract

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.

### Keywords

- Derangement polynomials
- Complex variable, Congruence
- Umbral calculus

### 2010 Mathematics Subject Classification

- 11B83
- 11A07
- 30C10

### References

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

## Related papers

- Benyattou, A., & Mihoubi, M. (2022). Note on some sequences having periods that divide (
*p*− 1) / (^{p}*p*− 1).*Notes on Number Theory and Discrete Mathematics*, 28(2), 234-239, DOI: 10.7546/nntdm.2022.28.2.234-239.

## 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.