Open Access

Degenerate Changhee-Genocchi numbers and polynomials

  • Byung Moon Kim1,
  • Lee-Chae Jang2Email author,
  • Wonjoo Kim3 and
  • Hyuck-In Kwon4
Journal of Inequalities and Applications20172017:294

https://doi.org/10.1186/s13660-017-1572-z

Received: 1 October 2017

Accepted: 17 November 2017

Published: 28 November 2017

Abstract

In this paper, we study some properties of degenerate Changhee-Genocchi numbers and polynomials and give some new identities of these polynomials and numbers which are derived from the generating function. In particular, we provide interesting identities related to the Changhee-Genocchi polynomials of the second kind and Changhee-Genocchi numbers of the second kind.

Keywords

Changhee numbersGenocchi numbersdegenerate Changhee-Genocchi numbers

MSC

05A1005A1911B6811S80

1 Introduction

Carlitz first introduced the concept of degenerate numbers and polynomials which are related to Bernoulli and Euler numbers and polynomials (see [1, 2]). After Carlitz introduced the degenerate polynomials, many researchers studied the degenerate polynomials associated with special polynomials in various areas (see [320]).

Recently, Kim and Kim gave same new and interesting identities of Changhee numbers and polynomials which are derived from the non-linear differential equation (see [9]). These identities and technical method are very useful for studying some problems which are related to mathematical physics.

As is known, the Genocchi polynomials are defined by the generating function to be
$$ \begin{aligned} \frac{2t}{e^{t}+1}e^{xt} = \sum _{n=0}^{\infty}G_{n}(x) \frac{t^{n}}{n!}\quad \bigl(\text{see [1--25]}\bigr). \end{aligned} $$
(1.1)
When \(x=0\), \(G_{n}=G_{n}(0)\) are called the Genocchi numbers. From (1.1), we note that
$$ \begin{aligned} G_{0}(x) =0,\qquad E_{n}(x) = \frac{G_{n+1}(x)}{n+1}\quad (n \geq0)\quad \bigl(\text{see [16]}\bigr). \end{aligned} $$
Here, \(E_{n}(x)\) are ordinary Euler polynomials which are given by the generating function to be
$$ \begin{aligned} \frac{2}{e^{t}+1}e^{xt} = \sum _{n=0}^{\infty}E_{n}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.2)
Recently, the Changhee polynomials have been defined by the generating function to be
$$ \begin{aligned} \frac{2}{t+2}(1+t)^{x} = \sum_{n=0}^{\infty}\mathit{Ch}_{n}(x) \frac{t^{n}}{n!}\quad \bigl(\text{see [9, 11--17]}\bigr). \end{aligned} $$
(1.3)
When \(x=0\), \(\mathit{Ch}_{n}=\mathit{Ch}_{n}(0)\) are called the Changhee numbers. From (1.3), we note that
$$ \begin{aligned}[b] \sum_{n=0}^{\infty}\mathit{Ch}_{n}(x) \frac{t^{n}}{n!}&= \frac{2}{e^{\log(1+t)}+1}e^{\log(1+t)x} \\ &= \sum_{k=0}^{\infty}E_{k}(x) \frac{1}{k!} \bigl(\log(1+t) \bigr)^{k} \\ &=\sum_{n=0}^{\infty}\Biggl( \sum _{k=0}^{n} E_{k}(x) S_{1}(n,k) \Biggr) \frac{t^{n}}{n!}, \end{aligned} $$
(1.4)
where \(S_{1}(n,k)\) is called the Stirling number of the first kind. From (1.4), we note that
$$ \begin{aligned} \mathit{Ch}_{n}(x) = \sum _{k=0}^{n} E_{k}(x) S_{1}(n,k)\quad (n \geq0) \quad\bigl(\text{see [9]}\bigr). \end{aligned} $$
(1.5)
As is well known, the Bernoulli numbers of the second kind are defined by the generating function to be
$$ \begin{aligned} \frac{t}{\log(1+t)} = \sum _{n=0}^{\infty}b_{n} \frac{t^{n}}{n!}\quad \bigl(\text{see} \text{[24]}\bigr). \end{aligned} $$
(1.6)
From (1.6), we note that
$$ \begin{aligned} \biggl( \frac{t}{\log(1+t)} \biggr)^{r} (1+t)^{x-1} = \sum_{n=0}^{\infty}B_{n}^{(n-r+1)}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.7)
When \(x=1\) and \(r=1\), we get
$$ \begin{aligned} b_{n} = B_{n}^{(n)}(1)\quad (n \geq0), \end{aligned} $$
where \(B_{n}^{(r)}(x)\) are the higher-order Bernoulli polynomials which are defined by the generating function to be
$$ \begin{aligned} \biggl( \frac{t}{e^{t}-1} \biggr)^{r} e^{xt} = \sum_{n=0}^{\infty}B_{n}^{(r)}(x) \frac{t^{n}}{n!}\quad\bigl(\text{see [6, 7, 11]}\bigr). \end{aligned} $$
In [17], Changhee-Genocchi polynomials are defined by the generating function to be
$$ \begin{aligned} \frac{2 \log(1+t)}{2+t} (1+t)^{x} = \sum_{n=0}^{\infty}\mathit{CG}_{n}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.8)
From (1.8), we note that
$$ \begin{aligned}[b] \frac{2 \log(1+t)}{2+t} (1+t)^{x} &= \frac{2\log(1+t)}{e^{\log(1+t)}+1} e^{x \log(1+t)} \\ &= \sum_{k=0}^{\infty}G_{k}(x) \frac{1}{k!} \bigl(\log(1+t) \bigr)^{k} \\ &= \sum_{n=0}^{\infty}\Biggl( \sum _{k=0}^{n} G_{k}(x) S_{1}(n,k) \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(1.9)
By (1.8) and (1.9), we get
$$ \begin{aligned} \mathit{CG}_{n}(x) =\sum _{k=0}^{n} G_{k}(x) S_{1}(n,k) \quad\bigl(\text{see [17, 18]}\bigr). \end{aligned} $$
(1.10)
Now, we consider the modified Changhee-Genocchi polynomials which are given by
$$ \begin{aligned} \frac{2t}{2+t}(1+t)^{x} = \sum_{n=0}^{\infty}\mathit{CG}_{n}^{*} (x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.11)
Now, we observe that
$$ \begin{aligned}[b] \frac{2t}{2+t}(1+t)^{x} &= \frac{2t}{e^{\log(1+t)}+1}e^{x\log(1+t)} \\ &= t \sum_{k=0}^{\infty}E_{k}(x) \frac{1}{k!} \bigl(\log(1+t) \bigr)^{k} \\ &= t \sum_{n=0}^{\infty}\Biggl( \sum _{k=0}^{n} E_{k}(x) S_{1}(n,k) \Biggr) \frac {t^{n}}{n!}. \end{aligned} $$
(1.12)
From (1.11), it is easy to show that \(\mathit{CG}_{0}^{*}(x) =0\). Thus, by (1.11) and (1.12), we get
$$ \begin{aligned} \frac{\mathit{CG}_{n+1}^{*} (x)}{n+1} = \sum _{k=0}^{n} E_{k}(x) S_{1}(n,k)\quad (n \geq0). \end{aligned} $$
(1.13)
As is known, the degenerate Euler polynomials were defined by Carlitz [1, 2] to be
$$ \begin{aligned} \frac{2}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac {x}{\lambda}} = \sum_{n=0}^{\infty}\mathcal{E}_{n,\lambda}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.14)
Now, we consider the following degenerate Genocchi polynomials which are derived from (1.14):
$$ \begin{aligned} \frac{2t}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac {x}{\lambda}} = \sum_{n=0}^{\infty}G_{n,\lambda}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.15)
Thus, by (1.15), we get
$$ \begin{aligned}[b] \sum_{n=0}^{\infty}\lim_{\lambda \rightarrow0} G_{n,\lambda}(x) \frac{t^{n}}{n!} &= \lim _{\lambda\rightarrow0} \frac{2t}{(1+\lambda t)^{\frac {1}{\lambda}}+1}(1+\lambda t)^{\frac{x}{\lambda}} \\ &= \frac{2t}{e^{t}+1}e^{xt} = \sum_{n=0}^{\infty}G_{n}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.16)
From (1.15) and (1.16)
$$ \begin{aligned} \lim_{\lambda\rightarrow0} G_{n, \lambda}(x)=G_{n}(x)\quad (n\geq0). \end{aligned} $$
(1.17)
It is not difficult to show that
$$ \begin{aligned}[b] \frac{2}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac {x}{\lambda}} &= \frac{1}{t} \frac{2t}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac {x}{\lambda}} \\ &= \frac{1}{t} \sum_{n=0}^{\infty}G_{n,\lambda}(x) \frac{t^{n}}{n!} \\ &= \frac{1}{t} \sum_{n=1}^{\infty}G_{n,\lambda}(x) \frac{t^{n}}{n!} \\ &=\sum_{n=0}^{\infty}\frac{G_{n+1,\lambda}(x)}{n+1} \frac{t^{n}}{n!}. \end{aligned} $$
(1.18)
Therefore, by (1.14) and (1.18), for \(n \geq0\), we get
$$ \begin{aligned} \mathcal{E}_{n,\lambda}(x) = \frac{1}{n+1} G_{n+1,\lambda}(x). \end{aligned} $$
When \(x=0\), \(G_{n,\lambda}= G_{n,\lambda}(0)\) are called the degenerate Genocchi numbers. From (1.15), we have
$$ \begin{aligned}[b] 2t &= \Biggl( \sum _{l=0}^{\infty}G_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \bigl(1+ (1+\lambda t)^{\frac{1}{\lambda}} \bigr) \\ &=\sum_{n=0}^{\infty}G_{n,\lambda} \frac{t^{n}}{n!} + \Biggl( \sum_{l=0}^{\infty}G_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum _{m=0}^{\infty}(1|\lambda)_{m} \frac{t^{m}}{m!} \Biggr) \\ &=\sum_{n=0}^{\infty}\Biggl\{ G_{n,\lambda} + \sum_{l=0}^{n} {n \choose l} G_{l,\lambda} (1|\lambda)_{n-l} \Biggr\} \frac{t^{n}}{n!}, \end{aligned} $$
(1.19)
where \((x|\lambda)_{n} = x(x-\lambda)\cdots(x-(n-1)\lambda)\) (\(n \geq 1\)), \((x|\lambda)_{0}=1\).
By comparing the coefficients on the both sides of (1.19), we have
$$ \begin{aligned} G_{n,\lambda} + \sum _{l=0}^{n} {n \choose l} G_{l,\lambda} (1|\lambda)_{n-l}= \textstyle\begin{cases} 0&\text{if } n=0, \\ 2\delta_{n,1}&\text{if } n=1, \\ 0&\text{if } n>1 . \end{cases}\displaystyle \end{aligned} $$
Note that \(G_{0,\lambda}=0\).
In [17], the degenerate Changhee-Genocchi polynomials are defined by the generating function to be
$$ \begin{aligned} \frac{2\lambda\log(1+ \frac{1}{\lambda}\log (1+\lambda t) ) }{2\lambda+\log(1+\lambda t)} \bigl( 1+ \lambda^{-1}\log(1+\lambda t) \bigr)^{x} = \sum _{n=0}^{\infty}\mathit{CG}_{n,\lambda}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.20)
From (1.8) and (1.20), we note that
$$ \begin{aligned} \lim_{\lambda \rightarrow0} \mathit{CG}_{n,\lambda}(x) = \mathit{CG}_{n}(x)\quad (n \geq 0 )\quad\bigl( \text{see [17]}\bigr). \end{aligned} $$
(1.21)
The modified degenerate Changhee-Genocchi polynomials are considered by the generating function to be
$$ \begin{aligned} \frac{2t\lambda}{2\lambda+\log(1+\lambda t)} \bigl(1+ \lambda^{-1}\log (1+\lambda t) \bigr)^{x} = \sum _{n=0}^{\infty}\mathit{CG}_{n,\lambda}^{*}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.22)

Note that \(\lim_{\lambda\rightarrow0} \mathit{CG}_{n,\lambda}^{*}(x) = \mathit{CG}_{n}^{*}(x)\) (\(n \geq0\)).

As is known, the degenerate Changhee polynomials are given by
$$ \begin{aligned} \frac{2}{2+\lambda^{-1}\log(1+\lambda t)} \bigl( 1+ \lambda^{-1}\log (1+\lambda t) \bigr)^{x} = \sum _{n=0}^{\infty}\mathit{Ch}_{n,\lambda}(x) \frac {t^{n}}{n!} \quad\bigl(\text{see [8]}\bigr). \end{aligned} $$
(1.23)
From (1.22) and (1.23), we have
$$ \begin{aligned} \mathit{Ch}_{n,\lambda}(x) = \frac{1}{n+1} \mathit{CG}_{n+1,\lambda}^{*} (x)\quad (n \geq0). \end{aligned} $$
(1.24)
When \(x=0\), \(\mathit{CG}_{n,\lambda}^{*} = \mathit{CG}_{n,\lambda}^{*}(0)\) are called the modified degenerate Changhee-Genocchi numbers. From (1.22), for \(x=0\), we note that
$$ \begin{aligned}[b] 2t &= \bigl( 2+\lambda^{-1} \log(1+\lambda t) \bigr) \Biggl( \sum_{l=0}^{\infty}\mathit{CG}_{l,\lambda}^{*} \frac{t^{l}}{l!} \Biggr) \\ &= \Biggl( 2+ \sum_{k=1}^{\infty}\frac{(-1)^{k-1} \lambda^{k-1}}{k} t^{k} \Biggr) \Biggl( \sum _{l=0}^{\infty}\mathit{CG}_{l,\lambda}^{*} \frac{t^{l}}{l!} \Biggr) \\ &= \sum_{n=1}^{\infty}\Biggl( 2\mathit{CG}_{n,\lambda}^{*} + \sum_{k=1}^{n} {n \choose k} (k-1)! (-1)^{k-1} \lambda^{k-1} \mathit{CG}_{n-k,\lambda}^{*} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(1.25)
Comparing the coefficients on the both sides of (1.25), we have
$$ \begin{aligned} 2\mathit{CG}_{n,\lambda}^{*} + \sum _{k=1}^{n} {n \choose k} (k-1)! (-1)^{k-1} \lambda^{k-1} \mathit{CG}_{n-k,\lambda}^{*} = 2 \delta_{n,1}, \end{aligned} $$
(1.26)
where \(\delta_{n,k}\) is the Kronecker delta symbol. In the viewpoint of (1.8), we consider the following partially degenerate Changhee-Genocchi polynomials which are derived from (1.20):
$$ \begin{aligned} \frac{2\lambda\log(1+t)}{2\lambda+\log(1+\lambda t)} \bigl(1+\lambda ^{-1}\log(1+\lambda t) \bigr)^{x} = \sum _{n=0}^{\infty}\mathit{PCG}_{n,\lambda}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(1.27)
When \(x=0\), \(\mathit{PCG}_{n,\lambda}=\mathit{PCG}_{n,\lambda}(0)\) are called the partially degenerate Changhee-Genocchi numbers. Note that \(\lim_{\lambda \rightarrow0}\mathit{PCG}_{n,\lambda}(x) = \mathit{CG}_{n}(x)\) (\(n \geq0\)).
From (1.27), for \(x=0\), we note that
$$ \begin{aligned}[b] 2\log(1+t) &= \bigl( 2+ \lambda^{-1}\log(1+\lambda t) \bigr) \Biggl( \sum _{l=0}^{\infty}\mathit{PCG}_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \\ &=\sum_{n=1}^{\infty}\Biggl( 2\mathit{PCG}_{n,\lambda} + \sum_{k=1}^{n} (-1)^{k-1} \lambda^{k-1} (k-1)! {n \choose k} \mathit{PCG}_{n-k,\lambda} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(1.28)
Thus, by (1.28), we get
$$ \begin{aligned} 2\mathit{PCG}_{n,\lambda} + \sum _{k=1}^{n} (-1)^{k-1} \lambda^{k-1} (k-1)! {n \choose k} \mathit{PCG}_{n-k,\lambda} = 2 (-1)^{n-1} (n-1)!, \end{aligned} $$
(1.29)
where \(n=1,2,3,\ldots \) .

In this paper, we study the degenerate Changhee-Genocchi polynomials and numbers of the second kind which are different from previous degenerate Changhee-Genocchi polynomials and numbers. In addition, we give some new identities from our numbers and polynomials.

2 Changhee-Genocchi numbers and polynomials of the second kind

First, we consider the Changhee-Genocchi polynomials of the second kind which are given by the generating function to be
$$ \begin{aligned} \frac{2\log(1+t)}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda }}+1} \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}}= \sum_{n=0}^{\infty}J_{n,\lambda}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(2.1)
Thus, by (1.8) and (2.1), we get
$$ \begin{aligned}[b] \sum_{n=0}^{\infty}\lim_{\lambda\rightarrow0} J_{n,\lambda} (x) \frac {t^{n}}{n!} &= \frac{2\log(1+t)}{2+t} (1+t)^{x} = \sum_{n=0}^{\infty}\mathit{CG}_{n}(x) \frac{t^{n}}{n!}. \end{aligned} $$
(2.2)
By comparing the coefficient on the both sides, we get
$$ \begin{aligned} \mathit{CG}_{n}(x) = \lim _{\lambda\rightarrow0} J_{n,\lambda} (x)\quad (n \geq0). \end{aligned} $$
(2.3)
When \(x=0\), \(J_{n,\lambda}= J_{n,\lambda}(0)\) are called the Changhee-Genocchi numbers of the second kind.
From (2.1) and (2.3), we note that
$$\begin{aligned}& \frac{2\log(1+t)}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda}}+1 } \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}} \\& \quad= \bigl(\log(1+t) \bigr) \Biggl( \sum_{l=0}^{\infty}\mathcal {E}_{l,\lambda}(x) \frac{1}{l!} \bigl( \log(1+t) \bigr)^{l} \Biggr) \\& \quad= \bigl(\log(1+t) \bigr) \Biggl( \sum_{k=0}^{\infty}\Biggl( \sum_{l=0}^{k} \mathcal{E}_{l,\lambda}(x) S_{1}(k,l) \Biggr) \frac{t^{k}}{k!} \Biggr) \\& \quad= \Biggl( \sum_{m=1}^{\infty}\frac{(-1)^{m-1}}{m} t^{m} \Biggr) \Biggl( \sum _{k=0}^{\infty}\Biggl( \sum_{l=0}^{k} \mathcal{E}_{l,\lambda}(x) S_{1}(k,l) \Biggr) \frac{t^{k}}{k!} \Biggr) \\& \quad= \sum_{n=1}^{\infty}\Biggl( \sum _{k=0}^{n-1} \sum_{l=0}^{k} \frac {(-1)^{n-k-1}n! \mathcal{E}_{l,\lambda}(x)}{(n-k) k!} S_{1}(k,l) \Biggr) \frac{t^{n}}{n!} \end{aligned}$$
(2.4)
and
$$ \begin{aligned}[b] &\frac{2\log(1+t)}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda}}+1 } \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}} \\ &\quad= \sum_{l=0}^{\infty}\mathcal{E}_{l,\lambda}(x) \frac{1}{l!} \bigl( \log (1+t) \bigr)^{l+1} \\ &\quad= \sum_{l=0}^{\infty}\mathcal{E}_{l,\lambda}(x) (l+1) \frac{1}{(l+1)!} \bigl( \log(1+t) \bigr)^{l+1} \\ &\quad= \sum_{l=0}^{\infty}\mathcal{E}_{l,\lambda}(x) (l+1) \sum_{n=l+1}^{\infty}S_{1}(n,l+1) \frac{t^{n}}{n!} \\ &\quad= \sum_{n=1}^{\infty}\Biggl( \sum _{l=0}^{n-1} \mathcal{E}_{l,\lambda }(x) (l+1) S_{1}(n,l+1) \Biggr) \frac{t^{n}}{n!}, \end{aligned} $$
(2.5)
where \(S_{1}(n,k)\) is the Stirling number of the first kind. By (2.1), (2.4) and (2.5), we obtain the following theorem.

Theorem 2.1

For \(n \geq1\), we have
$$ \begin{aligned} J_{n,\lambda}(x) = \sum _{k=0}^{n-1} \sum_{l=0}^{k} \frac {(-1)^{n-k-1}n!}{(n-k) k!} \mathcal{E}_{l,\lambda}(x)S_{1}(k,l) \end{aligned} $$
and
$$ \begin{aligned} J_{n,\lambda}(x) = \sum _{l=0}^{n-1} \mathcal{E}_{l,\lambda}(x) (l+1) S_{1}(n,l+1) . \end{aligned} $$
By replacing t by \(e^{t}-1\) in (2.1), we get
$$ \begin{aligned}[b] \frac{2t}{(1+\lambda t)^{\frac{1}{\lambda}}+1} (1+\lambda t)^{\frac {x}{\lambda}} &= \sum_{k=0}^{\infty}J_{k,\lambda}(x) \frac{1}{k!} \bigl(e^{t}-1 \bigr)^{k} \\ &= \sum_{k=0}^{\infty}J_{k,\lambda}(x) \sum _{n=k}^{\infty}S_{2}(n,k) \frac {t^{n}}{n!} \\ &= \sum_{n=0}^{\infty}\Biggl( \sum _{k=0}^{n} J_{k,\lambda}(x) S_{2}(n,k) \Biggr) \frac{t^{n}}{n!}, \end{aligned} $$
(2.6)
where \(S_{2}(n,k)\) is the Stirling number of the second kind. Therefore, by (1.15) and (2.6), we obtain the following theorem.

Theorem 2.2

For \(n \geq0\), we have
$$ \begin{aligned} G_{n,\lambda}(x) = \sum _{k=0}^{n} J_{k,\lambda}(x) S_{2}(n,k). \end{aligned} $$
From (2.1), we have
$$ \begin{aligned}[b] 2\log(1+t)&= \Biggl( \sum _{k=0}^{\infty}J_{k,\lambda} \frac{t^{k}}{k!} \Biggr) \bigl( \bigl(1+ \lambda \log(1+t) \bigr)^{\frac{1}{\lambda}}+1 \bigr) \\ &= \sum_{n=1}^{\infty}J_{n,\lambda} \frac{t^{n}}{n!} + \Biggl( \sum_{n=1}^{\infty}J_{n,\lambda} \frac{t^{n}}{n!} \Biggr) \Biggl( \sum _{l=0}^{\infty}(1|\lambda)_{l} \frac{1}{l!} \bigl( \log(1+t) \bigr)^{l} \Biggr) \\ &= \sum_{n=1}^{\infty}J_{n,\lambda} \frac{t^{n}}{n!} + \Biggl( \sum_{n=1}^{\infty}J_{n,\lambda} \frac{t^{n}}{n!} \Biggr) \Biggl( \sum _{m=0}^{\infty}\Biggl( \sum_{l=0}^{m} (1|\lambda)_{l} S_{1}(m,l) \Biggr) \frac {t^{m}}{m!} \Biggr) \\ &= \sum_{n=1}^{\infty}J_{n,\lambda} \frac{t^{n}}{n!} + \sum_{n=1}^{\infty}\Biggl( \sum_{m=0}^{n-1} \sum _{l=0}^{m} {n \choose m} (1| \lambda)_{l} S_{1}(m,l) J_{n-m,\lambda} \Biggr) \frac{t^{n}}{n!} \\ &= \sum_{n=1}^{\infty}\Biggl\{ J_{n,\lambda} + \sum_{m=0}^{n-1} \sum _{l=0}^{m} {n \choose m} (1|\lambda )_{l} S_{1}(m,l) J_{n-m,\lambda} \Biggr\} \frac{t^{n}}{n!}. \end{aligned} $$
(2.7)
It is easy to show that
$$ \begin{aligned} 2\log(1+t) = 2\sum _{n=1}^{\infty}\frac{(-1)^{n-1}}{n} t^{n} = 2 \sum _{n=1}^{\infty}(-1)^{n-1} (n-1)! \frac{t^{n}}{n!}. \end{aligned} $$
(2.8)
Therefore, by (2.7) and (2.8), we obtain the following theorem.

Theorem 2.3

For \(n \geq1\), we have
$$ \begin{aligned} J_{n,\lambda} + \sum_{m=0}^{n-1} \sum_{l=0}^{m} {n \choose m} (1| \lambda )_{l} S_{1}(m,l) J_{n-m,\lambda} = 2(-1)^{n-1}(n-1)!. \end{aligned} $$
The degenerate Changhee polynomials of the second kind are also defined by the generating function to be
$$ \begin{aligned} \frac{2}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda}}+1 } \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}} = \sum_{n=0}^{\infty}C_{n,\lambda}(x) \frac{t^{n}}{n!}\quad\bigl(\text{see [8]}\bigr). \end{aligned} $$
(2.9)
Now, we observe that
$$\begin{aligned}& \frac{2}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda}}+1 } \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}} \\& \quad= \frac{1}{t} \biggl( \frac{t}{\log(1+t)} \biggr) \biggl( \frac{2\log (1+t)}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda}}+1} \bigl(1+\lambda\log(1+t) \bigr)^{\frac{x}{\lambda}} \biggr) \\& \quad= \frac{1}{t} \Biggl( \sum_{l=0}^{\infty}b_{l} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum _{m=1}^{\infty}J_{m,\lambda}(x) \frac{t^{m}}{m!} \Biggr) \\& \quad= \Biggl( \sum_{l=0}^{\infty}b_{l} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty}\frac{J_{m+1,\lambda}(x)}{m+1} \frac{t^{m}}{m!} \Biggr) \\& \quad= \sum_{n=0}^{\infty}\Biggl( \sum _{m=0}^{n} {n \choose m} \frac {J_{m+1,\lambda}(x)}{m+1} b_{n-m} \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(2.10)

By comparing the coefficients on the both sides of (2.9) and (2.10), we obtain the following theorem.

Theorem 2.4

For \(n \geq0\), we have
$$ \begin{aligned} C_{n,\lambda}(x) = \sum _{m=0}^{n} {n \choose m} \frac{J_{m+1,\lambda }(x)}{m+1} b_{n-m}. \end{aligned} $$
From (2.1), we easily note that
$$ \begin{aligned}[b] \sum_{n=0}^{\infty}J_{n,\lambda}(x) \frac{t^{n}}{n!} &= \frac{2\log(1+t)}{ (1+\lambda\log(1+t) )^{\frac{1}{\lambda }}+1 } \bigl(1+\lambda \log(1+t) \bigr)^{\frac{x}{\lambda}} \\ &= \Biggl( \sum_{l=0}^{\infty}J_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty}\biggl( \frac{x}{\lambda} \biggr)_{m} \lambda^{m} \frac {1}{m!} \bigl(\log(1+t) \bigr)^{m} \Biggr) \\ &= \Biggl( \sum_{m=0}^{\infty}(x| \lambda)_{m} \sum_{k=m}^{\infty}S_{1}(k,m) \frac{t^{k}}{k!} \Biggr) \Biggl( \sum _{l=0}^{\infty}J_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \\ &= \Biggl( \sum_{k=0}^{\infty}\Biggl( \sum _{m=0}^{k} (x|\lambda)_{m} S_{1}(k,m) \Biggr) \frac{t^{k}}{k!} \Biggr) \Biggl( \sum _{l=0}^{\infty}J_{l,\lambda} \frac{t^{l}}{l!} \Biggr) \\ &=\sum_{n=0}^{\infty}\Biggl( \sum _{k=0}^{n} \sum_{m=0}^{k} {n \choose k} (x|\lambda)_{m} S_{1}(k,m) J_{n-k,\lambda} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(2.11)
Therefore, by comparing the coefficients on the both sides of (2.11), we obtain the following theorem.

Theorem 2.5

For \(n \geq0\), we have
$$ \begin{aligned} J_{n,\lambda}(x) = \sum _{k=0}^{n} \sum_{m=0}^{k} {n \choose k} (x|\lambda )_{m} S_{1}(k,m) J_{n-k,\lambda}. \end{aligned} $$

3 Conclusions

Kwon et al. [17] introduced the degenerate Changhee-Genocchi polynomials and numbers. In this study, we defined the modified degenerate Changhee-Genocchi polynomials and numbers (see (1.22)) and obtained an interesting identity (1.26) of the modified degenerate Changhee-Genocchi numbers. Secondly, we defined the partially degenerate Changhee-Genocchi polynomials and numbers (see (1.27)). We obtained a useful identity (1.29) of the partially degenerate Changhee-Genocchi numbers (1.29). Finally, we defined the Changhee-Genocchi polynomials of the second kind (see (2.1)). We provided useful identities related to the Changhee-Genocchi polynomials of the second kind and the degenerate Euler polynomials (see Theorem 2.1). Furthermore, we obtained some interesting identities of the Changhee-Genocchi polynomials of the second kind (see Theorem 2.2, Theorem 2.3, Theorem 2.4 and Theorem 2.5).

Declarations

Acknowledgements

The authors would like to express their gratitude for the valuable comments and suggestions of referees.

Authors’ contributions

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

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mechanical System Engineering, Dongguk University
(2)
Graduate School of Education, Konkuk University
(3)
Department of Applied Mathematics, Kyunghee University
(4)
Department of Mathematics, Kwangwoon University

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