# Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### Combinatorial Numbers

## 2. Changhee Polynomials of Negative Order

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

#### Computation Formula for Changhee Polynomials of Negative Order

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

## 3. Partial Derivative of the Generating Function $\mathit{H}(\mathit{t},\mathit{x},-\mathit{k})$

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

## 4. Integral Representations for Negative Order Changhee Polynomials

#### 4.1. Riemann Integral Representation for Negative Order Changhee Polynomials

**Theorem**

**9.**

**Proof.**

#### 4.2. p-Adic Integral Representations for Negative Order Changhee Polynomials

## 5. Identities and Relations

**Theorem**

**10.**

**Corollary**

**6.**

**Theorem**

**11.**

**Corollary**

**7.**

**Corollary**

**8.**

**Theorem**

**12.**

**Theorem**

**13.**

**Theorem**

**14.**

**Theorem**

**15.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{C}{\mathit{h}}_{\mathit{n}}^{(-\mathit{k})}$ | $\mathit{C}{\mathit{h}}_{\mathit{n}}^{(-\mathit{k})}\left(\mathit{x}\right)$ | |
---|---|---|

0 | 1 | 1 |

1 | $\frac{k}{2}$ | $x+\frac{k}{2}$ |

2 | $\frac{{k}^{2}-k}{4}$ | ${x}^{2}+(k-1)x+\frac{{k}^{2}-k}{4}$ |

3 | $\frac{{k}^{3}-3{k}^{2}+4k}{8}$ | ${x}^{3}+3\left(\frac{k}{2}-1\right){x}^{2}+\left(\frac{3{k}^{2}-9k+8}{4}\right)x+\frac{{k}^{3}-3{k}^{2}+4k}{8}$ |

4 | $\frac{{k}^{4}-6{k}^{3}+19{k}^{2}-28k}{16}$ | ${x}^{4}+(2k-6){x}^{3}+\left(\frac{3}{2}{k}^{2}-6k+11\right){x}^{2}+\left(\frac{{k}^{3}-3{k}^{2}+4k}{2}\right)x+\left(\frac{{k}^{4}-6{k}^{3}+19{k}^{2}-28k}{16}.\right)$ |

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Kim, D.; Simsek, Y.; So, J.S.
Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials. *Symmetry* **2020**, *12*, 9.
https://doi.org/10.3390/sym12010009

**AMA Style**

Kim D, Simsek Y, So JS.
Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials. *Symmetry*. 2020; 12(1):9.
https://doi.org/10.3390/sym12010009

**Chicago/Turabian Style**

Kim, Daeyeoul, Yilmaz Simsek, and Ji Suk So.
2020. "Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials" *Symmetry* 12, no. 1: 9.
https://doi.org/10.3390/sym12010009