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.