Discrete Dynamics in Nature and Society

Volume 2018, Article ID 9520269, 5 pages

https://doi.org/10.1155/2018/9520269

## On Carlitz’s Type Modified Degenerate -Changhee Polynomials and Numbers

^{1}Department of Mechanical System Engineering, Dongguk University, Gyeongju 780-714, Republic of Korea^{2}Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea^{3}Department of Applied Mathematics, Kyung Hee University, Seoul 02447, Republic of Korea^{4}Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Lee-Chae Jang; rk.ca.kuknok@gnajcl

Received 17 January 2018; Accepted 22 March 2018; Published 6 June 2018

Academic Editor: J. R. Torregrosa

Copyright © 2018 Byung Moon Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recently, Dolgy-Jang-Kwon-Kim introduced Carlitz’s type -Changhee polynomials. In this paper, we define Carlitz’s type modified degenerate -Changhee polynomials and investigate some interesting identities of these polynomials.

#### 1. Introduction

Let be a prime number with . Throughout this paper, , , and denote the ring of -adic integers, the field of -adic rational numbers, and the completion of the algebraic closure of , respectively. The -adic norm is normalized as . Let be an indeterminate in such that . The -analogue of number is defined as . As is well known, the Euler polynomials are defined by the generating function to be(see [1–5]).

When , are called the Euler numbers.

Recall that Carlitz considered the -analogue of Euler numbers which are given by the recurrence relation as follows:with the usual convention about replacing by , and that he also considered -Euler polynomials which are defined by(see [1–5]).

Let be the space of continuous -valued functions on . For , the fermionic -adic -integrals on are defined by Kim to bewhere (see [1, 3, 4, 6–14]). From (2), he derived the following formula for Carlitz’s -Euler numbers.

The Changhee polynomials are defined by the generating function to be(see [15–23]).

In [16, 24], the authors (2017) obtained that(see [15, 16, 18–23]), where is the Stirling numbers of the first kind and is the Stirling numbers of the second kind as follows:(see [2, 10, 25–27]).

The degenerate Euler polynomials are defined by the generating function to be(see [3, 4]).

The degenerate -Euler polynomials are defined by the generating function to be(see [3, 4]).

In [2, 26–28], Kim et al. (2017) defined the degenerate Stirling numbers of the second kind as follows:where and . In [15], by using the fermionic -adic -integral on , the authors defined Carlitz’s type -Changhee polynomials as follows:(see [1, 2, 14, 15, 24, 29–31]).

In this paper, we define Carlitz’s type modified degenerate -Changhee polynomials and investigate some interesting identities of these polynomials.

#### 2. Carlitz’s Type Modified Degenerate -Changhee Polynomials

In this section, we assume that with and . From (4) and (6), we note that

In the viewpoint of (12) and (14), Carlitz’s type modified degenerate -Changhee polynomials are defined byWe observe thatFrom (15) and (16), we getThus, by (17), we get the following theorem.

Theorem 1. *For , one has*

Note that

Replacing by in (20), we observe that

From (15) and (21), we get the following theorem.

Theorem 2. *For , one has*

Replacing by in (15), we getNote thatFrom (23) and (24), we get the following theorem.

Theorem 3. *For , one has*

When , are called Carlitz’s type modified degenerate -Changhee number. We also observe that

From (26), we get

By (27), we get the following theorem.

Theorem 4. *For , one has*

#### 3. Results and Discussions

This study was to define the modified degenerate -Changhee polynomials in (15). Theorem 1 is an interesting identity between the modified degenerate -Changhee polynomials and the -Euler polynomials. Theorem 2 is that the modified degenerate -Changhee polynomials is represented by a sum of products of the degenerate Stirling numbers of the second kind, the Stirling numbers of the first kind, and -Changhee polynomials. Theorem 3 is an identity between the degenerate -Euler polynomials and the modified degenerate -Changhee polynomials. Theorem 4 is an identity between the modified degenerate -Changhee polynomials and the -Euler polynomials. In the future, we will study to define and to investigate the higher-order modified degenerate -Changhee polynomials (see [3, 11]) and to investigate the symmetric identities of the modified degenerate -Changhee polynomials (see [3, 11]).

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

#### Acknowledgments

The authors would like to express their gratitude for the valuable comments and suggestions of Professor Dae San Kim.

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