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

Authors: Won Joo Kim; Lee-Chae Jang; Byung Moon Kim
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper, we study Carlitz’s typeq-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.


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## Body

## 1. Introduction and Preliminaries

Letp 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 q∈Cp, we normally assume 1-qp<p-1/p-1, so that qx=exp⁡(xlog⁡q) for xp≥1. The q-extension of x is defined as [x]q=1-qx/1-q for q≠1 and x for q=1.As is well known, Carlitz’sq-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 f∈UD(Zp, the p-adic q-integral on Zp is defined by Kim to be(2)Iqf=∫Zpfxdμqx=limN→∞1pNq∑x=0pN-1fxqx(see [3–6]). From (2), we note that(3)qnIqfn-Iqf=qq-1∑j=0n-1qjfj+qq-1log⁡q∑j=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-1log⁡qf′0(see [7]). Kim et al. [8] defined the q-Daehee polynomials by the generating function to be(5)q-1+q-1/log⁡qlog⁡1+tq-1+qt1+tx=∑n=0∞Dn,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/log⁡qlog⁡1+tq-1+qt1+tx.In [9–12], we recall that the Daehee polynomials are given by the generating function to be(7)∫Zp1+tx+ydμ0y=log⁡1+tt1+tx=∑n=0∞Dnxtnn!,and the q-Bernoulli polynomials are given by the generating function to be(8)∫Zpex+ytdμqy=q-1+q-1/log⁡qtqet-1=∑n=0∞Bn,qxtnn!.When x=0, Dn=Dn(0) are called the Daehee numbers and Bn,q=Bn,q(0), (n≥0), are called the q-Bernoulli numbers. Kim [13] proved that Carlitz’s q-Bernoulli polynomials can be represented by the p-adic q-integral on Zp:(9)∫Zpex+yqtdμqy=∑n=0∞βn,qxtnn!.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 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 Typeq-Daehee Polynomials

Lett∈Cp 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=0∞dn,qxtnn!,When x=0,dn,q=dn,q(0) are called Carlitz’s type q-Daehee numbers.Theorem 1 (see [18], Witt’s formula).
Letn≥0; we have(11)dn,q=∫Zpx+yqmdμqy.Kim [19] 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-1⋯x-n+1=∑k=0nS1n,kxk,n≥1and S2(n,k) is the Stirling numbers of the second kind as follows:(15)xn=∑l=0nS2n,lxl,n≥1.LetSn 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+t∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqdqw1⋯wn-1y=limN→∞1pNqw1⋯wn-1∑y=0pN-1qw1⋯wn-1y×1+t∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiq=limN→∞1wnpNqw1⋯wn-1∑m=0wn-1∑y=0pN-1qw1⋯wn-1m+wny×1+t∏i=1n-1wim+wny+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqFrom (16), we have(17)1w1⋯wn-1q∑k1=0w1-1∑k2=0w2-1⋯∑kn-1=0wn-1-1q∑i=1n-1∏n≥j≥1,j≠iwjki×∫Zp1+t∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqdqw1⋯wn-1y=limN→∞⁡1w1⋯wn-1wnpNq∑k1=0w1-1∑k2=0w2-1⋯∑kn-1=0wn-1-1∑m=0wn-1∑y=0pN-1q∑i=1n-1∏n≥j≥1,j≠iwjki×qw1⋯wn-1m+wny1+t∏i=1n-1wim+wny+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiq.As this expression is an invariant under any permutationσ∈Sn, we have the following theorem.Theorem 2.
Forn≥1,w1,⋯wn∈N, the expressions(18)1wσ1⋯wσn-1q∑k1=0wσ1-1∑k2=0wσ2-1⋯∑kn-1=0wσn-1-1q∑i=1n-1∏n≥j≥1,j≠iwσjki×∫Zp1+t∏i=1n-1wσiy+∏i=1nwσix+wσn∑i=1n-1∏n≥j≥1,j≠iwσjkiqdqwσ1⋯wσn-1yare the same for any σ∈Sn.We observe that(19)∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiq=∏i=1n-1wiqy+wnx+∑i=1n-1wnwiqw1⋯wn-1From (26) and Theorem 1, we note that(20)∫Zp1+t∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqdqw1⋯wn-1y=∑m=0∞∫Zp∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqmdqw1⋯wn-1ytmm!=∑m=0∞∑k=0mS1m,k∫Zp∏i=1n-1wiy+∏i=1nwix+wn∑i=1n-1∏n≥j≥1,j≠iwjkiqkdqw1⋯wn-1ytmm!=∑m=0∞∑k=0mS1m,k∏i=1n-1wiqk∑j=1kS2k,j×∫Zpy+wnx+∑i=1n-1wnwiqw1⋯wn-1kdqw1⋯wn-1ytmm!=∑m=0∞∑k=0mS1m,k∏i=1n-1wiqk∑j=1kS2k,jdk,qw1⋯wn-1wnx+∑i=1n-1wnwi.Therefore, by Theorem 2 and (20), we obtain the following theorem.Theorem 3.
Forn≥1,w1,⋯wn∈N, the expressions(21)∑k1=0wσ1-1∑k2=0wσ2-1⋯∑kn-1=0wσn-1-1q∑i=1n-1∏n≥j≥1,j≠iwσjki×∑m=0∞∑k=0mS1m,k∏i=1n-1wσiqk∑j=1kS2k,jdk,qwσ1⋯wσn-1wσnx+∑i=1n-1wσnwσiare the same for any σ∈Sn.We observe that(22)y+wnx+∑i=1n-1wnwikiq=1-qw1⋯wn-1y+w1⋯wnx+wn∑i=1n-1∏n≥j≥1,j≠iwjki1-qw1⋯wn-1=wnqw1⋯wn-1q∑i=1n-1∏n≥j≥1,j≠iwjkiqwn×q∑i=1n-1∏n≥j≥1,j≠iwjkiy+wnxqw1⋯wn-1.From (22), we note that(23)
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.By (24), we get(24)w1⋯wn-1qn-1∑k1=0w1-1∑k2=0w2-1⋯∑kn-1=0wn-1-1q∑i=1n-1∏n≥j≥1,j≠iwjki×∫Zpy+wnx+∑i=1n-1∏n≥j≥1,j≠iwjkiqw1⋯wn-1ndqw1⋯wn-1y=∑m=0nnmw1⋯wn-1qn-1wnn-m∑l=0mS2m,ldl,qw1⋯wn-1wnxTn,qwnw1⋯wn-1∣m,where(25)Tn,qwnw1⋯wn-1∣m∑k1=0w1-1∑k2=0w2-1⋯∑kn-1=0wn-1-1qm+1∑i=1n-1∏n≥j≥1,j≠iwjki∑i=1n-1∏n≥j≥1,j≠iwjkiqwnm-1.As this expression is an invariant under any permutationσ∈Sn, we have the following theorem.Theorem 4.
Forn≥1,w1,⋯wn∈N, the expressions(26)∑m=0nnmwσ1⋯wσn-1qn-1wσnn-m×∑l=0mS2m,ldl,qwσ1⋯wσn-1wnxTn,qwσnwσ1⋯wσn-1∣mare the same for any σ∈Sn.



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*Source: 1890489-2019-06-19.xml*