# k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- ${(x)}_{n+1,k}=(x+nk){(x)}_{n,k}.$
- (ii)
- ${(1)}_{n,1}=n!;\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(\frac{1}{2})}_{n,1}=\frac{(2n-1)!!}{{2}^{n}};\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(\frac{3}{2})}_{n,1}=\frac{(2n+1)!!}{{2}^{n}}.$
- (iii)
- ${(x)}_{n,1}=\frac{\mathsf{\Gamma}(x+n)}{\mathsf{\Gamma}(x)},\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}\mathsf{\Gamma}(x)\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{Gamma}\phantom{\rule{4.pt}{0ex}}\mathrm{function}\phantom{\rule{4.pt}{0ex}}\mathrm{defined}\phantom{\rule{4.pt}{0ex}}\mathrm{by}{\int}_{0}^{\infty}{e}^{-t}{t}^{x-1}dt.$
- (iv)
- ${(1)}_{n,2}=(2n-1)!!;\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(2)}_{n,2}=(2n)!!;\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(3)}_{n,2}=(2n+1)!!;\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(4)}_{n,2}=\frac{(2n+2)!!}{2}.$

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. The Solutions of Non-Homogeneous k-Hypergeometric Equations

**Lemma**

**1**

**.**If the series $\sum _{n=0}^{\infty}}{u}_{n$ satisfies the following condition:

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Li, S.; Dong, Y.
*k*-Hypergeometric Series Solutions to One Type of Non-Homogeneous *k*-Hypergeometric Equations. *Symmetry* **2019**, *11*, 262.
https://doi.org/10.3390/sym11020262

**AMA Style**

Li S, Dong Y.
*k*-Hypergeometric Series Solutions to One Type of Non-Homogeneous *k*-Hypergeometric Equations. *Symmetry*. 2019; 11(2):262.
https://doi.org/10.3390/sym11020262

**Chicago/Turabian Style**

Li, Shengfeng, and Yi Dong.
2019. "*k*-Hypergeometric Series Solutions to One Type of Non-Homogeneous *k*-Hypergeometric Equations" *Symmetry* 11, no. 2: 262.
https://doi.org/10.3390/sym11020262