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.
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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 [3–20]).
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
When \(x=0\), \(G_{n}=G_{n}(0)\) are called the Genocchi numbers. From (1.1), we note that
Here, \(E_{n}(x)\) are ordinary Euler polynomials which are given by the generating function to be
Recently, the Changhee polynomials have been defined by the generating function to be
When \(x=0\), \(\mathit{Ch}_{n}=\mathit{Ch}_{n}(0)\) are called the Changhee numbers. From (1.3), we note that
where \(S_{1}(n,k)\) is called the Stirling number of the first kind. From (1.4), we note that
As is well known, the Bernoulli numbers of the second kind are defined by the generating function to be
From (1.6), we note that
When \(x=1\) and \(r=1\), we get
where \(B_{n}^{(r)}(x)\) are the higher-order Bernoulli polynomials which are defined by the generating function to be
In [17], Changhee-Genocchi polynomials are defined by the generating function to be
From (1.8), we note that
Now, we consider the modified Changhee-Genocchi polynomials which are given by
Now, we observe that
From (1.11), it is easy to show that \(\mathit{CG}_{0}^{*}(x) =0\). Thus, by (1.11) and (1.12), we get
As is known, the degenerate Euler polynomials were defined by Carlitz [1, 2] to be
Now, we consider the following degenerate Genocchi polynomials which are derived from (1.14):
Thus, by (1.15), we get
It is not difficult to show that
Therefore, by (1.14) and (1.18), for \(n \geq0\), we get
When \(x=0\), \(G_{n,\lambda}= G_{n,\lambda}(0)\) are called the degenerate Genocchi numbers. From (1.15), we have
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
Note that \(G_{0,\lambda}=0\).
In [17], the degenerate Changhee-Genocchi polynomials are defined by the generating function to be
From (1.8) and (1.20), we note that
The modified degenerate Changhee-Genocchi polynomials are considered by the generating function to be
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
From (1.22) and (1.23), we have
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
Comparing the coefficients on the both sides of (1.25), we have
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):
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
Thus, by (1.28), we get
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
Thus, by (1.8) and (2.1), we get
By comparing the coefficient on the both sides, we get
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
and
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
and
By replacing t by \(e^{t}-1\) in (2.1), we get
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
From (2.1), we have
It is easy to show that
Therefore, by (2.7) and (2.8), we obtain the following theorem.
Theorem 2.3
For \(n \geq1\), we have
The degenerate Changhee polynomials of the second kind are also defined by the generating function to be
Now, we observe that
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
From (2.1), we easily note that
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
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).
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Kim, B.M., Jang, LC., Kim, W. et al. Degenerate Changhee-Genocchi numbers and polynomials. J Inequal Appl 2017, 294 (2017). https://doi.org/10.1186/s13660-017-1572-z
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DOI: https://doi.org/10.1186/s13660-017-1572-z