Research Article | Open Access

Volume 2019 |Article ID 1890489 | https://doi.org/10.1155/2019/1890489

Won Joo Kim, Lee-Chae Jang, Byung Moon Kim, "On Symmetric Identities of Carlitz’s Type -Daehee Polynomials", Discrete Dynamics in Nature and Society, vol. 2019, Article ID 1890489, 5 pages, 2019. https://doi.org/10.1155/2019/1890489

# On Symmetric Identities of Carlitz’s Type -Daehee Polynomials

Academic Editor: Alicia Cordero
Accepted08 Apr 2019
Published19 Jun 2019

#### Abstract

In this paper, we study Carlitz’s type -Daehee polynomials and investigate the symmetric identities for them by using the -adic -integral on under the symmetry group of degree .

#### 1. Introduction and Preliminaries

Let be a fixed prime number. Throughout this paper, , , and will 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 . If , we normally assume , so that for . The -extension of is defined as for and for .

As is well known, Carlitz’s -Bernoulli numbers are defined bywith the usual convention about replacing by (see [1, 2]). Let be the space of uniformly differentiable functions on . For , the -adic -integral on is defined by Kim to be(see [36]). From (2), we note thatwhere and . In particular, if we take , then we have(see [7]). Kim et al. [8] defined the -Daehee polynomials by the generating function to beWhen , are called the Daehee numbers with -parameter. By (4), we getIn [912], we recall that the Daehee polynomials are given by the generating function to beand the -Bernoulli polynomials are given by the generating function to beWhen , are called the Daehee numbers and , , are called the -Bernoulli numbers. Kim [13] proved that Carlitz’s -Bernoulli polynomials can be represented by the -adic -integral on :Kim-Kim-Jang [14] gave symmetric identities for degenerate Berstein and degenerate Euler polynomials and also many mathematical researchers studied symmetry identities of various polynomials (see [1, 1517]). In this paper, we consider Carlitz’s type -Daehee polynomials and investigate the symmetry identities for them by using the -adic -integral on under the symmetry group of degree .

#### 2. Symmetry Identities for Carlitz’s Type -Daehee Polynomials

Let with . From (6), we consider Carlitz’s type -Daehee polynomials can be represented by the -adic -integral on :When are called Carlitz’s type -Daehee numbers.

Theorem 1 (see [18], Witt’s formula). Let ; we have

Kim [19] obtained thatandwhere is the Stirling numbers of the first kind as follows:and is the Stirling numbers of the second kind as follows:

Let be the symmetry group of degree . For positive integers , we consider the following integral equation for the -adic -integral on .From (16), we have

As this expression is an invariant under any permutation , we have the following theorem.

Theorem 2. For , the expressionsare the same for any .

We observe that

From (26) and Theorem 1, we note thatTherefore, by Theorem 2 and (20), we obtain the following theorem.

Theorem 3. For , the expressionsare the same for any .

We observe that

From (22), we note thatBy (24), we getwhere

As this expression is an invariant under any permutation , we have the following theorem.

Theorem 4. For , the expressionsare the same for any .

#### Data Availability

The numerical simulation data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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