DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2019/1890489 1890489 Research Article On Symmetric Identities of Carlitz’s Type q-Daehee Polynomials Kim Won Joo 1 https://orcid.org/0000-0001-5317-1725 Jang Lee-Chae 2 Kim Byung Moon 3 Cordero Alicia 1 Department of Applied Mathematics Kyunghee University Seoul Republic of Korea khu.ac.kr 2 Graduate School of Education Konkuk University Seoul 05029 Republic of Korea konkuk.ac.kr 3 Department of Mechanical System Engineering Dongguk University Gyeongju Republic of Korea dongguk.edu 2019 1962019 2019 19 02 2019 08 04 2019 1962019 2019 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.

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

1. Introduction and Preliminaries

Let p be a fixed prime number. Throughout this paper, Zp, Qp, and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp, respectively. The p-adic norm is normalized as pp=1/p. If qCp, we normally assume 1-qp<p-1/p-1, so that qx=exp(xlogq) for xp1. The q-extension of x is defined as [x]q=1-qx/1-q for q1 and x for q=1.

As is well known, Carlitz’s q-Bernoulli numbers are defined by(1)β0,q=1,qqβq+1k-βk,q=1,ifk=1,0,ifk>1,with the usual convention about replacing βqk by βk,q (see [1, 2]). Let UD(Zp) be the space of uniformly differentiable functions on Zp. For fUD(Zp, the p-adic q-integral on Zp is defined by Kim to be(2)Iqf=Zpfxdμqx=limN1pNqx=0pN-1fxqx(see ). From (2), we note that(3)qnIqfn-Iqf=qq-1j=0n-1qjfj+qq-1logqj=0n-1fjqj,where fn(x)=f(x+n) and f(j)=d/dxf(x)x=j. In particular, if we take n=1, then we have(4)qIqf1-Iqf=qq-1f0+qq-1logqf0(see ). Kim et al.  defined the q-Daehee polynomials by the generating function to be(5)q-1+q-1/logqlog1+tq-1+qt1+tx=n=0Dn,qxtnn!.When x=0, Dn,q=Dn,q(0) are called the Daehee numbers with q-parameter. By (4), we get(6)Zp1+tx+ydμqy=q-1+q-1/logqlog1+tq-1+qt1+tx.In , we recall that the Daehee polynomials are given by the generating function to be(7)Zp1+tx+ydμ0y=log1+tt1+tx=n=0Dnxtnn!,and the q-Bernoulli polynomials are given by the generating function to be(8)Zpex+ytdμqy=q-1+q-1/logqtqet-1=n=0Bn,qxtnn!.When x=0, Dn=Dn(0) are called the Daehee numbers and Bn,q=Bn,q(0), (n0), are called the q-Bernoulli numbers. Kim  proved that Carlitz’s q-Bernoulli polynomials can be represented by the p-adic q-integral on Zp:(9)Zpex+yqtdμqy=n=0βn,qxtnn!.Kim-Kim-Jang  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 q-Daehee polynomials and investigate the symmetry identities for them by using the p-adic q-integral on Zp under the symmetry group of degree n.

2. Symmetry Identities for Carlitz’s Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M66"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Daehee Polynomials

Let tCp with tp<p-1/p-1. From (6), we consider Carlitz’s type q-Daehee polynomials can be represented by the p-adic q-integral on Zp:(10)Zp1+tx+yqdμqy=n=0dn,qxtnn!,When x=0,dn,q=dn,q(0) are called Carlitz’s type q-Daehee numbers.

Theorem 1 (see [<xref ref-type="bibr" rid="B10">18</xref>], Witt’s formula).

Let n0; we have(11)dn,q=Zpx+yqmdμqy.

Kim  obtained that(12)dn,q=k=0nS1n,kβk,qxand(13)βn,q=k=0nS2n,kdk,qxwhere S1(n,k) is the Stirling numbers of the first kind as follows:(14)x0=1,xn=xx-1x-n+1=k=0nS1n,kxk,n1and S2(n,k) is the Stirling numbers of the second kind as follows:(15)xn=l=0nS2n,lxl,n1.

Let Sn be the symmetry group of degree n. For positive integers w1,w2,,wn, we consider the following integral equation for the p-adic q-integral on Zp.(16)Zp1+ti=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiqdqw1wn-1y=limN1pNqw1wn-1y=0pN-1qw1wn-1y×1+ti=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiq=limN1wnpNqw1wn-1m=0wn-1y=0pN-1qw1wn-1m+wny×1+ti=1n-1wim+wny+i=1nwix+wni=1n-1nj1,jiwjkiqFrom (16), we have(17)1w1wn-1qk1=0w1-1k2=0w2-1kn-1=0wn-1-1qi=1n-1nj1,jiwjki×Zp1+ti=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiqdqw1wn-1y=limN1w1wn-1wnpNqk1=0w1-1k2=0w2-1kn-1=0wn-1-1m=0wn-1y=0pN-1qi=1n-1nj1,jiwjki×qw1wn-1m+wny1+ti=1n-1wim+wny+i=1nwix+wni=1n-1nj1,jiwjkiq.

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

Theorem 2.

For n1,w1,wnN, the expressions(18)1wσ1wσn-1qk1=0wσ1-1k2=0wσ2-1kn-1=0wσn-1-1qi=1n-1nj1,jiwσjki×Zp1+ti=1n-1wσiy+i=1nwσix+wσni=1n-1nj1,jiwσjkiqdqwσ1wσn-1yare the same for any σSn.

We observe that(19)i=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiq=i=1n-1wiqy+wnx+i=1n-1wnwiqw1wn-1

From (26) and Theorem 1, we note that(20)Zp1+ti=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiqdqw1wn-1y=m=0Zpi=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiqmdqw1wn-1ytmm!=m=0k=0mS1m,kZpi=1n-1wiy+i=1nwix+wni=1n-1nj1,jiwjkiqkdqw1wn-1ytmm!=m=0k=0mS1m,ki=1n-1wiqkj=1kS2k,j×Zpy+wnx+i=1n-1wnwiqw1wn-1kdqw1wn-1ytmm!=m=0k=0mS1m,ki=1n-1wiqkj=1kS2k,jdk,qw1wn-1wnx+i=1n-1wnwi.Therefore, by Theorem 2 and (20), we obtain the following theorem.

Theorem 3.

For n1,w1,wnN, the expressions(21)k1=0wσ1-1k2=0wσ2-1kn-1=0wσn-1-1qi=1n-1nj1,jiwσjki×m=0k=0mS1m,ki=1n-1wσiqkj=1kS2k,jdk,qwσ1wσn-1wσnx+i=1n-1wσnwσiare the same for any σSn.

We observe that(22)y+wnx+i=1n-1wnwikiq=1-qw1wn-1y+w1wnx+wni=1n-1nj1,jiwjki1-qw1wn-1=wnqw1wn-1qi=1n-1nj1,jiwjkiqwn×qi=1n-1nj1,jiwjkiy+wnxqw1wn-1.

From (22), we note that (23) Z p y + w n x + i = 1 n - 1 n j 1 , j i w j k i q w 1 w n - 1 n d q w 1 w n - 1 y = m = 0 n n m w n q w 1 w n - 1 q n - m i = 1 n - 1 n j 1 , j i w j k i q w n n - m × q m i = 1 n - 1 n j 1 , j i w j k i Z p y + w n x q w 1 w n - 1 m d μ q w 1 w n - 1 y = m = 0 n n m w n q w 1 w n - 1 q n - m i = 1 n - 1 n j 1 , j i w j k i q w n n - m × q m i = 1 n - 1 n j 1 , j i w j k i l = 0 m S 2 m , l Z p y + w n x q w 1 w n - 1 l d μ q w 1 w n - 1 y = m = 0 n n m w n q w 1 w n - 1 q n - m i = 1 n - 1 n j 1 , j i w j k i q w n n - m × q m i = 1 n - 1 n j 1 , j i w j k i l = 0 m S 2 m , l d l , q w 1 w n - 1 w n x . By (24), we get(24)w1wn-1qn-1k1=0w1-1k2=0w2-1kn-1=0wn-1-1qi=1n-1nj1,jiwjki×Zpy+wnx+i=1n-1nj1,jiwjkiqw1wn-1ndqw1wn-1y=m=0nnmw1wn-1qn-1wnn-ml=0mS2m,ldl,qw1wn-1wnxTn,qwnw1wn-1m,where(25)Tn,qwnw1wn-1mk1=0w1-1k2=0w2-1kn-1=0wn-1-1qm+1i=1n-1nj1,jiwjkii=1n-1nj1,jiwjkiqwnm-1.

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

Theorem 4.

For n1,w1,wnN, the expressions(26)m=0nnmwσ1wσn-1qn-1wσnn-m×l=0mS2m,ldl,qwσ1wσn-1wnxTn,qwσnwσ1wσn-1mare the same for any σSn.

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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