# On Carlitz’s Type Modified Degenerateq-Changhee Polynomials and Numbers

Authors: Byung Moon Kim; Lee-Chae Jang; Wonjoo Kim; Hyuck-In Kwon
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Recently, Dolgy-Jang-Kwon-Kim introduced Carlitz’s typeq-Changhee polynomials. In this paper, we define Carlitz’s type modified degenerate q-Changhee polynomials and investigate some interesting identities of these polynomials.


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## Body

## 1. Introduction

Letp be a prime number with p≡1(mod⁡2). Throughout this paper, Zp, Qp, and Cp 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 ·p is normalized as pp=1/p. Let q be an indeterminate in Cp such that 1-qp<p-1/(p-1). The q-analogue of number x is defined as [x]q=(qx-1)/(q-1). As is well known, the Euler polynomials are defined by the generating function to be(1)2et+1ext=∑n=0∞Enxtnn!(see [1–5]).Whenx=0, En=En(0)(n≥0) are called the Euler numbers.Recall that Carlitz considered theq-analogue of Euler numbers which are given by the recurrence relation as follows:(2)E0,q=1,qqEq+1n+En,q=2q,ifn=0,0,ifn≥1with the usual convention about replacing Eqn by En,q, and that he also considered q-Euler polynomials which are defined by(3)En,qx=∑l=0nnlxqn-lqlxEl,q(see [1–5]).LetC(Zp) be the space of continuous Cp-valued functions on Zp. For f∈C(Zp), the fermionic p-adic q-integrals on Zp are defined by Kim to be(4)I-qf=∫Zpfxdμ-qx=limN→∞⁡1pN-q∑x=0pN-1fx-qx,where [x]-q=(1-(-q)x)/(1+q) (see [1, 3, 4, 6–14]). From (2), he derived the following formula for Carlitz’s q-Euler numbers.The Changhee polynomials are defined by the generating function to be(5)22+t1+tx=∑n=0∞Chnxtnn!(see [15–23]).In [16, 24], the authors (2017) obtained that(6)Enx=∑k=0nS2n,kChkx,(7)Chnx=∑k=0nS1n,kEkx,n≥0(see [15, 16, 18–23]), where S1(n,k) is the Stirling numbers of the first kind and S2(n,k) is the Stirling numbers of the second kind as follows:(8)xn=xx-1⋯x-n+1=∑k=0nS1n,kxk,et-1m=m!∑k=m∞S2k,mtkk!,n≥0(see [2, 10, 25–27]).The degenerate Euler polynomials are defined by the generating function to be(9)21+λt1/λ+11+λtx/λ=∑n=0∞En,λxtnn!(see [3, 4]).The degenerateq-Euler polynomials are defined by the generating function to be(10)∫Zp1+λtx+yq/λdμ-qy=∑n=0∞En,q,λxtnn!(see [3, 4]).In [2, 26–28], Kim et al. (2017) defined the degenerate Stirling numbers of the second kind as follows:(11)1k!1+λt1/λ-1k=∑n=k∞S2,λn,ktnn!,where k∈N∪{0} and λ∈R. In [15], by using the fermionic p-adic q-integral on Zp, the authors defined Carlitz’s type q-Changhee polynomials as follows:(12)∫Zp1+tx+yqdμ-qy=∑n=0∞Chn,qxtnn!(see [1, 2, 14, 15, 24, 29–31]).In this paper, we define Carlitz’s type modified degenerateq-Changhee polynomials and investigate some interesting identities of these polynomials.


## 2. Carlitz’s Type Modified Degenerateq-Changhee Polynomials

In this section, we assume thatt∈Cp with |t|p<p-1/(p-1) and λp<p-1/(p-1). From (4) and (6), we note that(13)∫Zp1+tx+ydμ-1y=22+t1+tx=∑n=0∞Chnxtnn!,(14)1+t=elog⁡1+t=limλ→0⁡1+λlog⁡1+t1/λ.In the viewpoint of (12) and (14), Carlitz’s type modified degenerate q-Changhee polynomials are defined by(15)∫Zp1+λlog⁡1+tx+yq/λdμ-qy=∑n=0∞Chn,q,λxtnn!.We observe that(16)1+λlog⁡1+tx+yq/λ=∑k=0∞1λx+yqkλklog⁡1+tk=∑k=0∞1λx+yqkλk1k!log⁡1+tk=∑k=0∞1λx+yqkλk∑m=k∞S1m,ktmm!=∑m=0∞∑k=0mS1m,k1λx+yqkλktmm!=∑m=0∞∑k=0mS1m,k∑l=0k1λlx+yqlS1k,lλktmm!=∑m=0∞∑k=0m∑l=0kS1m,kS1k,lλk-lx+yqltmm!.From (15) and (16), we get(17)∑n=0∞Chn,q,λxtnn!=∫Zp1+λlog⁡1+tx+yq/λdμ-qy=∑n=0∞∑k=0n∑l=0kS1n,kS1k,lλk-l∫Zpx+yqldμ-qytnn!=∑n=0∞∑k=0n∑l=0kS1n,kS1k,lλk-lEl,qxtnn!.Thus, by (17), we get the following theorem.Theorem 1.
Forn≥0, one has(18)Chn,q,λx=∑k=0n∑l=0kS1n,kS1k,lλk-lEl,qx.Note that(19)limλ→0⁡Chn,q,λx=limλ→0⁡∑k=0n∑l=0kS1n,kλk-lEl,qx=∑k=0nSn,kEk,qx=Chn,qx,(20)∫Zp1+tx+yqdμ-qy=∑n=0∞Chn,qxtnn!.Replacingt by (1+λlog⁡(1+t))1/λ-1 in (20), we observe that(21)∫Zp1+λlog⁡1+t1/λx+yqdμ-qy=∑m=0∞Chm,qx1m!1+λlog⁡1+t1/λ-1m=∑m=0∞Chm,qx∑k=m∞S2,λk,mlog⁡1+tkk!=∑k=0∞∑m=0kChm,qxS2,λk,m∑n=k∞S1n,ktnn!=∑n=0∞∑k=0n∑m=0kChn,qxS2,λk,mS1n,ktnn!.From (15) and (21), we get the following theorem.Theorem 2.
Forn≥0, one has(22)Chn,q,λx=∑k=0n∑m=0kChm,qxS2,λk,mS1n,k.Replacingt by et-1 in (15), we get(23)∫Zp1+λt1/λx+yqdμ-qy=∑k=0∞Chk,q,λx1k!et-1k=∑k=0∞Chk,q,λx∑n=k∞S2n,ktnn!=∑n=0∞∑k=0nChk,q,λxS2n,ktnn!.Note that(24)∫Zp1+λt1/λx+yqdμ-qy=∑n=0∞En,q,λxtnn!.From (23) and (24), we get the following theorem.Theorem 3.
Forn≥0, one has(25)En,q,λx=∑k=0∞Chk,q,λxS2n,k.Whenx=0, Chn,q,λ=Chn,q,λ(0) are called Carlitz’s type modified degenerate q-Changhee number. We also observe that(26)1+λlog⁡1+t1/λ+1-1x+yq/λ=∑k=0∞x+yqλk1+λlog⁡1+t1/λ-1k=∑k=0∞x+yqλk1k!1+λlog⁡1+t1/λ-1k=∑k=0∞x+yqλk∑m=k∞S2,λm,k1m!log⁡1+tm=∑m=0∞∑k=0mx+yqλkS2,λm,k∑n=m∞S1n,mtnn!=∑n=0∞∑m=0n∑k=0mS2,λm,k∑l=0kS1k,lλ-lx+yqltnn!=∑n=0∞∑m=0n∑k=0m∑l=0kS2,λm,kS1k,lλ-lx+yqltnn!.From (26), we get(27)∑n=0∞Chn,q,λxtnn!=∫Zp1+λlog⁡1+t1/λ+1-1x+yq/λdμ-qy=∑n=0∞∑m=0n∑k=0m∑l=0kS2,λm,kS1k,lλ-l∫Zpx+yqldμ-qytnn!.By (27), we get the following theorem.Theorem 4.
Forn≥0, one has(28)Chn,q,λx=∑m=0n∑k=0m∑l=0kS2,λm,kSk,lλ-lEl,qx.


## 3. Results and Discussions

This study was to define the modified degenerateq-Changhee polynomials in (15). Theorem 1 is an interesting identity between the modified degenerate q-Changhee polynomials and the q-Euler polynomials. Theorem 2 is that the modified degenerate q-Changhee polynomials is represented by a sum of products of the degenerate Stirling numbers of the second kind, the Stirling numbers of the first kind, and q-Changhee polynomials. Theorem 3 is an identity between the degenerate q-Euler polynomials and the modified degenerate q-Changhee polynomials. Theorem 4 is an identity between the modified degenerate q-Changhee polynomials and the q-Euler polynomials. In the future, we will study to define and to investigate the higher-order modified degenerate q-Changhee polynomials (see [3, 11]) and to investigate the symmetric identities of the modified degenerate q-Changhee polynomials (see [3, 11]).



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*Source: 9520269-2018-06-06.xml*