On Symmetric Identities of Carlitz’s Type <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.58866 12.3952" width="8.58866pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg>-</span>Daehee Polynomials

Kim, Won Joo; Jang, Lee-Chae; Kim, Byung Moon

doi:https://doi.org/10.1155/2019/1890489

Discrete Dynamics in Nature and Society

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

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

On Symmetric Identities of Carlitz’s Type -Daehee Polynomials

Won Joo Kim,¹Lee-Chae Jang,²and Byung Moon Kim³

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 [3–6]). 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 [9–12], 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, 15–17]). 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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Copyright

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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