Abstract
Recently, Kim and Kim introduced some identities of degenerate Daehee numbers which are derived from nonlinear differential equations (see (Kim and Kim in J. Nonlinear Sci. Appl. 10:744-751, 2017)). From the viewpoint of inversion formula, we study the degenerate Daehee number arising from a nonlinear differential equation. In this paper, we obtain the explicit expression of degenerate Daehee numbers from the inversion formula of (Kim and Kim in J. Nonlinear Sci. Appl. 10:744-751, 2017) using the generating function and nonlinear differential equations.
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1 Introduction
The Daehee polynomials are defined by the generating function to be
For \(x=0\), \(D_{n}=D_{n}(0)\) are called the Daehee numbers.
In [1], Kim and Kim introduced the degenerate Daehee numbers which are given by the generating function:
For \(x=0\), \(D_{n,\lambda }=D_{n,\lambda }(0)\) are called the degenerate Daehee numbers.
We observe here that \(D_{n,\lambda } \longrightarrow D_{n}\) as \(\lambda \longrightarrow 0\).
The Stirling numbers of the first kind are given by
and the Stirling numbers of the first kind are defined by the generating function to be
Recently, many researchers have studied nonlinear differential equations arising from the generating functions of various special polynomials (see [1–6, 8–20]). They also investigated some identities and explicit expression of these polynomials from the solution of nonlinear differential equations. In [1], Kim and Kim have studied some results of degenerate Daehee numbers which are derived from nonlinear differential equations. From the viewpoint of the inversion formula, we study the degenerate Daehee number arising from a nonlinear differential equation. In this paper, by using the generating function and nonlinear differential equations, we deduce the explicit expression of degenerate Daehee numbers as the inversion formula of [1].
2 Some identities of degenerate Daehee numbers arising from nonlinear differential equations
Let
Then, by taking the derivative with respect to t of (2.1), we get
From (2.2), we get
From (2.3), we note that
Thus, by multiplying \((1+\lambda t)\) on both sides of (2.4), we get
From (2.6), we have
Multiplying \((1+\lambda t)\) on both sides of (2.7), we get
From (2.9), we have
Multiplying \((1+\lambda t)\) on both sides of (2.10), we get
Continuing this process, we get
Let us take the derivative on both sides of (2.13) with respect to t. Then we have
Multiplying \((1+\lambda t)\) on both sides of (2.14), we get
Then, by (2.3) and (2.15), we get
By replacing N by \(N+1\) in (2.13), we get
Comparing the coefficients on both sides of (2.16) and (2.17), we have
and
From (2.3) and (2.13), we have
By (2.20), we get
Thus, by (2.18) and (2.21), we have
and
For \(k=2\) in (2.19), we have
Then by (2.22), (2.23) and (2.24), we get
For \(k=3\) in (2.19), we have
Then by (2.25) and (2.26), we get
For \(k=4\) in (2.19), we have
Continuing this process, for \(2\leq k \leq N\), we have
Therefore, we obtain the following differential equations.
Theorem 2.1
Let \(N \in \mathbb{N}\). Then the differential equations
have a solution \(F=F(t)=\log (1+\frac{1}{\lambda }\log (1+\lambda t) )\), where
and
From (2.1), we easily get
From (2.31), we get
From (2.32), we get
and also
Here \(S_{1}(n,k)\) is the Stirling number of the first kind.
Thus, by (2.13) and (2.33), we get
By equation (2.37), we finally get the following theorem.
Theorem 2.2
For \(N=1,2,3,\ldots \) , and \(n=0,1,2,\ldots \) , we have
3 Conclusion
Kim and Kim have studied some identities of degenerate Daehee numbers which are derived from the generating function using nonlinear differential equation (see [1]). In this paper, from the viewpoint of the inversion formula to [1], we study the degenerate Daehee number arising from nonlinear differential equation. Therefore we obtain the inversion formula of degenerate Daehee numbers which are related to the some identities of those numbers. In Theorem 2.1, we get the solution of nonlinear differential equation arising from the generating function of the degenerate Daehee number. In Theorem 2.2, we have an explicit expression of the degenerate Daehee number from the result of Theorem 2.1 using the generating function and nonlinear differential equations.
References
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Acknowledgements
The authors would like to express their sincere gratitude to the editor, who gave us valuable comments to improve this paper. This paper was supported by Wonkwang University in 2017.
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Jang, GW., Kwon, J. & Lee, J.G. Some identities of degenerate Daehee numbers arising from nonlinear differential equation. Adv Differ Equ 2017, 206 (2017). https://doi.org/10.1186/s13662-017-1265-4
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DOI: https://doi.org/10.1186/s13662-017-1265-4