Discrete Dynamics in Nature and Society

Volume 2019, Article ID 1890489, 5 pages

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

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

^{1}Department of Applied Mathematics, Kyunghee University, Seoul, Republic of Korea^{2}Graduate School of Education, Konkuk University, Seoul 05029, Republic of Korea^{3}Department of Mechanical System Engineering, Dongguk University, Gyeongju, Republic of Korea

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

Received 19 February 2019; Accepted 8 April 2019; Published 19 June 2019

Academic Editor: Alicia Cordero

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.

#### 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.

#### References

- D. S. Kim and T. Kim, “Some identities of symmetry for
*q*-Bernoulli polynomials under symmetry group of degree*n*,”*Ars Combinatoria*, vol. 126, pp. 435–441, 2016. View at Google Scholar · View at MathSciNet - J.-W. Park, “On the
*q*-analogue of*λ*-Daehee polynomials,”*Journal of Computational Analysis and Applications*, vol. 19, no. 6, pp. 966–974, 2015. View at Google Scholar · View at MathSciNet - U. Duran, M. Acikgoz, and S. Araci, “Symmetric identities involving weighted
*q*-Genocchi polynomials under S4,”*Proceedings of the Jangjeon Mathematical Society. Memoirs of the Jangjeon Mathematical Society*, vol. 18, no. 4, pp. 455–465, 2015. View at Google Scholar · View at MathSciNet - L. Carlitz, “
*q*-Bernoulli numbers and polynomials,”*Duke Mathematical Journal*, vol. 18, no. 2, pp. 987–1000, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, D. V. Dolgy, and D. S. Kim, “Some identities of
*q*-Bernoulli polynomials under symmetry group*S*_{3},”*Journal of Nonlinear and Convex Analysis*, vol. 16, no. 9, pp. 1869–1880, 2015. View at Google Scholar · View at MathSciNet - J.-W. Park and S.-H. Rim, “On the modified
*q*-Bernoulli polynomials with weight,”*Proceedings of the Jangjeon Mathematical Society*, vol. 17, no. 2, pp. 231–236, 2014. View at Google Scholar · View at MathSciNet - D. V. Dolgy, T. Kim, S.-H. Rim, and S. H. Lee, “Symmetry identities for the generalized higher-order
*q*-Bernoulli polynomials under S3 arising from*p*-adic Volkenborn integral on Zp,”*Proceedings of the Jangjeon Mathematical Society*, vol. 17, no. 4, pp. 645–650, 2014. View at Google Scholar · View at MathSciNet - D. S. Kim, S. H. Lee, T. Mansour, and J.-J. Seo, “A note on
*q*-Daehee polynomials and numbers,”*Advanced Studies in Contemporary Mathematics*, vol. 24, no. 2, pp. 435–441, 2016. View at Google Scholar - B. S. El-Desouky, “New results on higher-order Daehee and Bernoulli numbers and polynomials,”
*Advances in Difference Equations*, vol. 32, p. 21, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - B. M. Kim, S. J. Yun, and J.-W. Park, “On a degenerate
*λ*-q-Daehee polynomials,”*Journal of Nonlinear Sciences and Applications*, vol. 9, no. 6, pp. 4607–4616, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - J.-W. Park, S.-H. Rim, and J. Kwon, “The twisted Daehee numbers and polynomials,”
*Advances in Difference Equations*, vol. 2014, p. 9, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - D. Rim, “Some identities for Carlitz’s type
*q*-Daehee polynomials,”*The Ramanujan Journal*, pp. 1–12, 2018. View at Google Scholar - T. Kim, “
*q*-Volkenborn integration,”*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at MathSciNet - T. Kim, D. S. Kim, and G.-W. Jang, “Symmetric identities for degenerate Berstein and degenerate Euler polynomials,”
*Symmery*, vol. 10, no. 6, p. 219, 2018. View at Google Scholar - T. Kim and D. S. Kim, “Identities of symmetry for degenerate Euler polynomials and alternating generalized falling factorial sums,”
*Iranian Journal of Science & Technology*, vol. 41, no. 4, pp. 939–949, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, H.-I. Kwon, T. Mansour, and S.-H. Rim, “Symmetric identities for the fully degenerate Bernoulli polynomials and degenerate Euler polynomials under symmetric group of degree
*n*,”*Utilitas Mathematica*, vol. 103, pp. 61–72, 2017. View at Google Scholar · View at MathSciNet - T. Kim, “Symmetric identities of degenerate Bernoulli polynomials,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 18, no. 4, pp. 593–599, 2015. View at Google Scholar · View at MathSciNet - T. Kim, D. V. Dolgy, L.-C. Jang, and H.-I. Kwon, “Some identities of degenerate
*q*-Euler polynomials under the symmetry group of degree*n*,”*Journal of Nonlinear Sciences and Applications*, vol. 9, no. 6, pp. 4707–4712, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - D. S. Kim, N. Lee, J. Na, and K. H. Park, “Abundant symmetry for higher-order Bernoulli polynomials (I),”
*Advanced Studies in Contemporary Mathematics*, vol. 23, no. 3, pp. 461–482, 2013. View at Google Scholar · View at MathSciNet