# On r-Central Incomplete and Complete Bell Polynomials

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- The first r elements are in different blocks,
- Any block of size i with no elements of the first r elements, can be colored with ${a}_{i}$ colors,
- Any block of size i with one element of the first r elements, can be colored with ${b}_{i}$ colors.

## 3. An Extended r-Central Complete and Incomplete Bell Polynomials

- The first r elements are in different blocks,
- Any block of (odd) size i with no elements of the first r elements, can be colored with ${a}_{i}$ colors,
- Any block of size i with one element of the first r elements, can be colored with ${b}_{i}$ colors.

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Corollary**

**4.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kim, D.S.; Dolgy, D.V.; Kim, D.; Kim, T. Some identities on r-central factorial numbers and r-central Bell polynomials. arXiv
**2019**, arXiv:1903.11689v1. [Google Scholar] - Riordan, J. Combinatorial Identities; John Wiley & Sons, Inc.: New York, NY, USA, 1968. [Google Scholar]
- Kim, T.; Kim, D.S.; Jang, G.-W.; Kwon, J. Extended central factorial polynomials of the second kind. Adv. Differ. Equ.
**2019**, 24. [Google Scholar] [CrossRef] - Kim, T.; Kim, D.S. Degenerate central Bell numbers and polynomials. Rev. Real Acad. Clenc. Exactas Fis. Nat. Ser. A Mat.
**2019**, 1–7. [Google Scholar] [CrossRef] - Zhang, W. Some identities involving the Euler and the central factorinal numbers. Fibonacci Quart.
**1998**, 36, 154–157. [Google Scholar] - Butzer, P.L.; Schmidt, M.; Stark, E.L.; Vogt, L. Central factorial numbers; their main properties and some applications. Numer. Funct. Anal. Optim.
**1989**, 10, 419–488. [Google Scholar] [CrossRef] - Carlitz, L.; Riordan, J. The divided central differences of zero. Canad. J. Math.
**1963**, 15, 94–100. [Google Scholar] [CrossRef] - Carlitz, L. Some remarks on the Bell numbers. Fibonacci Quart.
**1980**, 18, 66–73. [Google Scholar] - Charalambides, C.A. Central factorial numbers and related expansions. Fibonacci Quart.
**1981**, 19, 451–456. [Google Scholar] - Kim, T. A note on central factorial numbers. Proc. Jangjeon Math. Soc.
**2018**, 21, 575–588. [Google Scholar] - Kim, T.; Kim, D.S. A note on central Bell numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang)
**2019**, 27, 289–298. [Google Scholar] - Belbachir, H.; Djemmada, Y. On central Fubini-like numbers and polynomials. arXiv
**2018**, arXiv:1811.06734v1. [Google Scholar] - Krzywonos, N.; Alayont, F. Rook polynomials in higher dimensions. Stud. Summer Sch.
**2009**, 29. Available online: https://scholarworks.gvsu.edu/sss/29/ (accessed on 2 March 2019). - Duran, U.; Acikgoz, M.; Araci, S. On (q,r,w)-Stirling numbers of the second kind. J. Inequal. Spec. Funct.
**2018**, 9, 9–16. [Google Scholar] - Kim, T.; Yao, Y.; Kim, D.S.; Jang, G.-W. Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys.
**2018**, 25, 44–58. [Google Scholar] [CrossRef] - Pyo, S.-S. Degenerate Cauchy numbers and polynomials of the fourth kind. Adv. Stud. Contemp. Math. (Kyungshang)
**2018**, 28, 127–138. [Google Scholar] - Roman, S. The umbral calculus. In Pure and Applied Mathematics; Harcourt Brace Jovanovich: New York, NY, USA, 1984. [Google Scholar]
- Simsek, Y. Identities and relations related to combinatorial numbers and polynomials. Proc. Jangjeon Math. Soc.
**2017**, 20, 127–135. [Google Scholar] - Simsek, Y. Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang)
**2017**, 27, 199–212. [Google Scholar] - Mihoubi, M.; Rahmani, M. The partial r-Bell polynomials. Afr. Mat.
**2017**, 28, 1167–1183. [Google Scholar] [CrossRef] [Green Version] - Kim, T.; Kim, D.S.; Jang, G.-W. On central complete and incomplete Bell polynomials I. Symmetry
**2019**, 11, 288. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, D.S.; Kim, H.Y.; Kim, D.; Kim, T.
On *r*-Central Incomplete and Complete Bell Polynomials. *Symmetry* **2019**, *11*, 724.
https://doi.org/10.3390/sym11050724

**AMA Style**

Kim DS, Kim HY, Kim D, Kim T.
On *r*-Central Incomplete and Complete Bell Polynomials. *Symmetry*. 2019; 11(5):724.
https://doi.org/10.3390/sym11050724

**Chicago/Turabian Style**

Kim, Dae San, Han Young Kim, Dojin Kim, and Taekyun Kim.
2019. "On *r*-Central Incomplete and Complete Bell Polynomials" *Symmetry* 11, no. 5: 724.
https://doi.org/10.3390/sym11050724