Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2019 / Article

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
Received19 Feb 2019
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.

References

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Copyright © 2019 Won Joo 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.


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