# On Carlitz's Type Modified Degenerate g-Changhee Polynomials and Numbers.

1. Introduction

Let p be a prime number with p [equivalent to] 1(mod2). Throughout this paper, [Z.sub.p], [Q.sub.p], and [C.sub.p] denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of [Q.sub.p], respectively. The p-adic norm [[absolute value of *].sub.p] is normalized as [[absolute value of p].sub.p] = 1/p. Let q be an indeterminate in [C.sub.p] such that [[absolute value of 1 - q].sub.p] < [p.sup.-1/(p-1)]. The q-analogue of number x is defined as [[absolute value of x].sub.q] = ([q.sup.x] - 1)/(q- 1).As iswell known, the Euler polynomials are defined by the generating function to be

[2/[e.sup.t]+1] [e.sup.xt] = [[infinity].summation over (n=0)][E.sub.n](x) [t.sup.n]/n! (1)

(see [1-5]).

When x = 0, [E.sub.n] = [E.sub.n](0)(n [greater than or equal to] 0) are called the Euler numbers.

Recall that Carlitz considered the q-analogue of Euler numbers which are given by the recurrence relation as follows:

[mathematical expression not reproducible], (2)

with the usual convention about replacing [E.sup.n.sub.q] by [E.sub.n,q], and that he also considered q-Euler polynomials which are defined by

[mathematical expression not reproducible] (3)

(see [1-5]).

Let C([Z.sub.p]) be the space of continuous [C.sub.p]-valued functions on [Z.sub.p]. For f [member of] C([Z.sub.p]), the fermionic p-adic q-integrals on [Z.sub.p] are defined by Kim to be

[mathematical expression not reproducible] (4)

where [[x].sub.-q] = (1 - [(-q).sup.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

2/[2+t][(1+t).sup.x] = [[infinity].summation over (n=0)][Ch.sub.n](x)[t.sup.n]/n! (5)

(see [15-23]).

In [16, 24], the authors (2017) obtained that

[E.sub.n](x) = [n.summation over (k=0)][S.sub.2](n,k)[Ch.sub.k](x), (6)

[Ch.sub.n](x) = [member of][S.sub.1](n,k)[E.sub.k](x), (n [greater than or equal to] 0) (7)

(see [15, 16, 18-23]), where [S.sub.1](n, k) is the Stirling numbers of the first kind and [S.sub.2](n, k) is the Stirling numbers of the second kind as follows:

[(x).sub.n] = x(x-1) ... (x-n+1) = [n.summation over (k=0)][S.sub.1](n,k)[x.sup.k], [([e.sup.t]-1).sup.m] = m![[infinity].summation over (k=m)][S.sub.2](k,m)[t.sup.k]/k!, (n [greater than or equal to] 0) (8)

(see [2,10, 25-27]).

The degenerate Euler polynomials are defined by the generating function to be

2/[(1+[lambda]t).sup.1/[lambda]]+1 [(1+[lambda]t).sup.x/[lambda]] = [[infinity].summation over (n=0)][E.sub.n,[lambda]](x)[t.sup.n]/n! (9)

(see [3, 4]).

The degenerate q-Euler polynomials are defined by the generating function to be

[mathematical expression not reproducible] (10)

(see [3, 4]).

In [2, 26-28], Kim et al. (2017) defined the degenerate Stirling numbers of the second kind as follows:

1/k[([(1+[lambda]t).sup.1/[lambda]]-1).sup.k] = [[infinity].summation over (n=k)][S.sub.2,[lambda]](n,k)[t.sup.n]/n!, (11)

where k [member of] N [union]{0} and [lambda] n[member of] R. In [15], by using the fermionic p-adic q-integral on [Z.sub.p], the authors defined Carlitz's type q-Changhee polynomials as follows:

[mathematical expression not reproducible] (12)

(see [1, 2, 14, 15, 24, 29-31]).

In this paper, we define Carlitz's type modified degenerate q-Changhee polynomials and investigate some interesting identities of these polynomials.

2. Carlitz's Type Modified Degenerate g-Changhee Polynomials

In this section, we assume that t [member of] [C.sub.p] with [[absolute value of t].sub.p] < [p.sup.-1/(p-1)] and [[absolute value of [lambda]].sub.p] < [p.sup.-1/(p-1)]. From (4) and (6), we note that

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible]. (14)

In the viewpoint of (12) and (14), Carlitz's type modified degenerate q-Changhee polynomials are defined by

[mathematical expression not reproducible] (15)

We observe that

[mathematical expression not reproducible] (16)

From (15) and (16), we get

[mathematical expression not reproducible] (17)

Thus, by (17), we get the following theorem.

Theorem 1. For n [greater than or equal to] 0, one has

[Ch.sub.n,q,[lambda]](x) = [n.summation over (k=0)][k.summation over (l=0)][S.sub.1](n,k)[S.sub.1](k,l)[[lambda].sup.k- l][E.sup.l,q](x). (18)

Note that

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

Replacing t by [(1 + [lambda] log(1 + t)).sup.1/[lambda]] -1 in (20), we observe that

[mathematical expression not reproducible] (21)

From (15) and (21), we get the following theorem.

Theorem 2. For n [greater than or equal to] 0, one has

[Ch.sub.n,q,[lambda]](x) = [n.summation over (k=0)][k.summation over (m=0)][Ch.sub.m,q](x)[S.sub.2,[lambda]](k,m)[S.sub.1](n,k). (22)

Replacing t by [e.sup.t] -1 in (15), we get

[mathematical expression not reproducible] (23)

Note that

[mathematical expression not reproducible] (24)

From (23) and (24), we get the following theorem.

Theorem 3. For n [greater than or equal to] 0, one has

[E.sub.n,q,[lambda]](x) = [[infinity].summation over (k=0)][Ch.sub.k,q,[lambda]](x)[S.sub.2](n,k). (25)

When x = 0, [Ch.sub.n,q,[lambda]] = [Ch.sub.n,q,[lambda]](0) are called Carlitz's type modified degenerate q-Changhee number. We also observe that

[mathematical expression not reproducible] (26)

From (26), we get

[mathematical expression not reproducible] (27)

By (27), we get the following theorem.

Theorem 4. For n [greater than or equal to] 0, one has

[Ch.sub.n,q,[lambda]](x) = [n.summation over (m=0)][m.summation over (k=0)][k.summation over (l=0)][S.sub.2,[lambda]](m,k)[S.sub.k,l][[lambda].sup.-1][E.sup.l,q](x). (28)

3. Results and Discussions

This study was to define the modified degenerate q-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]).

https://doi.org/10.1155/2018/9520269

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to express their gratitude for the valuable comments and suggestions of Professor Dae San Kim.

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Byung Moon Kim, (1) Lee-Chae Jang (iD), (2) Wonjoo Kim, (3) and Hyuck-In Kwon (4)

(1) Department of Mechanical System Engineering, Dongguk University, Gyeongju 780-714, Republic of Korea

(2) Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea

(3) Department of Applied Mathematics, KyungHee University, Seoul 02447, Republic of Korea

(4) Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Lee-Chae Jang; lcjang@konkuk.ac.kr

Received 17 January 2018; Accepted 22 March 2018; Published 6 June 2018