Open Access

Some identities of degenerate Daehee numbers arising from nonlinear differential equation

Advances in Difference Equations20172017:206

https://doi.org/10.1186/s13662-017-1265-4

Received: 13 June 2017

Accepted: 6 July 2017

Published: 26 July 2017

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.

Keywords

differential equationsdegenerate Daehee numbersStirling numbers

MSC

11B6811S4011S80

1 Introduction

The Daehee polynomials are defined by the generating function to be
$$ \frac{\log (1+t)}{t}(1+t)^{x}=\sum _{n=0}^{\infty }D_{n}(x) \frac{t^{n}}{n!} $$
(1.1)
(see [27]).

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:
$$ \frac{\lambda \log (1+\frac{1}{\lambda }\log (1+\lambda t) )}{ \log (1+\lambda t)}=\sum_{n=0}^{\infty }D_{n,\lambda } \frac{t^{n}}{n!}. $$
(1.2)

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
$$ (x)_{n}= x(x-1)\cdots (x-n+1)=\sum ^{n} _{l=0}S_{1} (n,l)x^{l} \quad (x \geq 0), $$
(1.3)
and the Stirling numbers of the first kind are defined by the generating function to be
$$ \bigl( \log (x+1) \bigr) ^{n}=n!\sum _{l=n} ^{\infty }S_{1}(l,n)\frac{x ^{l}}{l!}\quad(n\geq 0) $$
(see [1, 3]).

Recently, many researchers have studied nonlinear differential equations arising from the generating functions of various special polynomials (see [16, 820]). 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
$$ \begin{aligned} F=F(t)=\log \biggl(1+\frac{1}{\lambda } \log (1+\lambda t) \biggr). \end{aligned} $$
(2.1)
Then, by taking the derivative with respect to t of (2.1), we get
$$\begin{aligned} F^{(1)}&=\frac{d}{dt}F(t)= \biggl(1+\frac{1}{\lambda }\log (1+ \lambda t) \biggr)^{-1}\frac{1}{1+\lambda t} \\ &=\frac{1}{1+\lambda t} e^{-\log (1+\frac{1}{\lambda }\log (1+ \lambda t) )} \\ &=\frac{1}{1+\lambda t}e^{-F}. \end{aligned}$$
(2.2)
From (2.2), we get
$$ \begin{aligned} e^{-F}=(1+\lambda t)F^{(1)}. \end{aligned} $$
(2.3)
From (2.3), we note that
$$ \begin{aligned} \bigl(-F^{(1)} \bigr)e^{-F}=\lambda F^{(1)}+(1+\lambda t)F^{(2)}. \end{aligned} $$
(2.4)
Thus, by multiplying \((1+\lambda t)\) on both sides of (2.4), we get
$$ \begin{aligned} (1+\lambda t)F^{(1)}e^{-F}=- \lambda (1+\lambda t)F^{(1)}-(1+\lambda t)^{2}F ^{(2)}. \end{aligned} $$
(2.5)
From (2.3) and (2.5), we get
$$ \begin{aligned} e^{-2F}=-\lambda (1+\lambda t)F^{(1)}-(1+\lambda t)^{2}F^{(2)}. \end{aligned} $$
(2.6)
From (2.6), we have
$$ \begin{aligned} -2F^{(1)}e^{-2F}=- \lambda^{2}F^{(1)}-3\lambda (1+\lambda t)F^{(2)}-(1+ \lambda t)^{2}F^{(3)}. \end{aligned} $$
(2.7)
Multiplying \((1+\lambda t)\) on both sides of (2.7), we get
$$\begin{aligned} 2!(1+\lambda t)F^{(1)}e^{-2F} &=(-1)^{2} \lambda^{2}(1+\lambda t)F^{(1)}+(-1)^{2} 3\lambda (1+\lambda t)^{2}F^{(2)} \\ &\quad {}+(-1)^{2}(1+\lambda t)^{3}F^{(3)}. \end{aligned}$$
(2.8)
From (2.3) and (2.8), we get
$$\begin{aligned} 2!e^{-3F} &=(-1)^{2}\lambda^{2}(1+\lambda t)F^{(1)}+ (-1)^{2} 3\lambda (1+\lambda t)^{2}F^{(2)} +(-1)^{2}(1+\lambda t)^{3}F^{(3)}. \end{aligned}$$
(2.9)
From (2.9), we have
$$\begin{aligned} 2!\bigl(-3F^{(1)}\bigr)e^{-3F} &=(-1)^{2} \lambda^{3}F^{(1)}+ (-1)^{2} 7\lambda ^{2}(1+\lambda t)F^{(2)} \\ &\quad {}+(-1)^{2} 6\lambda (1+\lambda t)^{2}F^{(3)}+(-1)^{2}(1+ \lambda t)^{3}F ^{(4)}. \end{aligned}$$
(2.10)
Multiplying \((1+\lambda t)\) on both sides of (2.10), we get
$$\begin{aligned} 3!(1+\lambda t)F^{(1)}e^{-3F} &=(-1)^{3} \lambda^{3}(1+\lambda t)F^{(1)}+(-1)^{3} 7 \lambda^{2}(1+\lambda t)^{2}F^{(2)} \\ &\quad {}+(-1)^{3} 6\lambda (1+\lambda t)^{3}F^{(3)}+(-1)^{3}(1+ \lambda t)^{4}F ^{(4)}. \end{aligned}$$
(2.11)
From (2.3) and (2.11), we get
$$\begin{aligned} 3!e^{-4F} &=(-1)^{3}\lambda^{3}(1+\lambda t)F^{(1)}+ (-1)^{3} 7\lambda ^{2}(1+\lambda t)^{2}F^{(2)} \\ &\quad {}+(-1)^{3} 6\lambda (1+\lambda t)^{3}F^{(3)}+(-1)^{3}(1+ \lambda t)^{4}F ^{(4)}. \end{aligned}$$
(2.12)
Continuing this process, we get
$$\begin{aligned} (N-1)!e^{-NF}=(-1)^{N-1}\sum_{k=1}^{N} \lambda^{N-k}(1+\lambda t)^{k} a _{k}(N)F^{(k)}. \end{aligned}$$
(2.13)
Let us take the derivative on both sides of (2.13) with respect to t. Then we have
$$\begin{aligned} (N-1)!\bigl(-N F^{(1)}\bigr)e^{-NF} &=(-1)^{N-1}\sum _{k=1}^{N}\lambda^{N-k}a_{k}(N) \bigl\{ k\lambda (1+\lambda t)^{k-1} \\ &\quad {}\times F^{(k)} +(1+\lambda t)^{k}F^{(k+1)} \bigr\} . \end{aligned}$$
(2.14)
Multiplying \((1+\lambda t)\) on both sides of (2.14), we get
$$\begin{aligned} N!(1+\lambda t)F^{(1)}e^{-NF} &=(-1)^{N}\sum _{k=1}^{N}\lambda^{N-k}a _{k}(N) \bigl\{ k\lambda (1+\lambda t)^{k} \\ &\quad {}\times F^{(k)} +(1+\lambda t)^{k+1}F^{(k+1)} \bigr\} . \end{aligned}$$
(2.15)
Then, by (2.3) and (2.15), we get
$$\begin{aligned} N!e^{-(N+1)F}&=(-1)^{N}\sum_{k=1}^{N} \lambda^{N-K}a_{k}(N) \bigl\{ k \lambda (1+\lambda t)^{k} \\ &\quad {}\times F^{(k)} +(1+\lambda t)^{k+1}F^{(k+1)} \bigr\} \\ &=(-1)^{N}\sum_{k=1}^{N} \lambda^{N-k+1}(1+\lambda t)^{k}ka_{k}(N)F^{(k)} \\ &\quad {}+(-1)^{N}\sum_{k=1}^{N} \lambda^{N-k}(1+\lambda t)^{k+1}a_{k}(N)F ^{(k+1)} \\ &=(-1)^{N}\sum_{k=1}^{N} \lambda^{N-k+1}(1+\lambda t)^{k}ka_{k}(N)F ^{(k)} \\ &\quad {}+(-1)^{N}\sum_{k=2}^{N+1} \lambda^{N-k+1}(1+\lambda t)^{k}a_{k-1}(N)F ^{(k)} \\ &=(-1)^{N}\lambda^{N}(1+\lambda t)a_{1}{(N)}F^{(1)}+(-1)^{N}(1+1 \lambda t)^{N+1} \\ &\quad {}\times a_{N}{(N)} F^{(N+1)}+(-1)^{N}\sum _{k=2}^{N}\lambda^{N-k+1}(1+ \lambda t)^{k} \\ &\quad {}\times \bigl(ka_{k}(N)+a_{k-1}(N) \bigr)F^{(k)}. \end{aligned}$$
(2.16)
By replacing N by \(N+1\) in (2.13), we get
$$\begin{aligned} N!e^{-(N+1)F} &=(-1)^{N}\sum_{k=1}^{N+1} \lambda^{N-k+1}(1+\lambda t)^{k} a_{k}(N+1)F^{(k)} \\ &=(-1)^{N}\lambda^{N}(1+\lambda t)a_{1}{(N+1)}F^{(1)}+(-1)^{N}(1+ \lambda t)^{N+1} \\ &\quad {}\times a_{N+1}{(N+1)}F^{(N+1)}+(-1)^{N}\sum _{k=2}^{N}\lambda^{N-k+1}(1+ \lambda t)^{k} \\ &\quad {}\times a_{k}{(N+1)}F^{k}. \end{aligned}$$
(2.17)
Comparing the coefficients on both sides of (2.16) and (2.17), we have
$$\begin{aligned} a_{1}(N+1)=a_{1}(N),\qquad a_{N+1}(N+1)=a_{N}(N), \end{aligned}$$
(2.18)
and
$$ a_{k}(N+1)=ka_{k}(N)+a_{k-1}(N), \quad \mbox{for } 2\leq k\leq N. $$
(2.19)
From (2.3) and (2.13), we have
$$ e^{-F}=(1+\lambda t)F^{(1)}=(1+\lambda t)a_{1}(1)F^{(1)}. $$
(2.20)
By (2.20), we get
$$ a_{1}(1)=1. $$
(2.21)
Thus, by (2.18) and (2.21), we have
$$ a_{1}(N+1)=a_{1}(N)=a_{1}(N-1)= \cdots =a_{1}(1)=1 $$
(2.22)
and
$$ a_{N+1}(N+1)=a_{N}(N)=a_{N-1}(N-1)= \cdots =a_{1}(1)=1. $$
(2.23)
For \(k=2\) in (2.19), we have
$$\begin{aligned} a_{2}(N+1) &=2a_{2}(N)+a_{1}(N) \\ &=2 \bigl(2a_{2}(N-1)+a_{1}(N-1) \bigr)+a_{1}(N) \\ &=2^{2}a_{2}(N-1)+2a_{1}(N-1)+a_{1}(N) \\ &=\cdots \\ &=2^{N-1}a_{2}(2)+2^{N-2}a_{1}(2)+\cdots +a_{1}(N). \end{aligned}$$
(2.24)
Then by (2.22), (2.23) and (2.24), we get
$$\begin{aligned} a_{2}(N+1) &=2^{N-1}a_{2}(2)+2^{N-2}a_{1}(2)+ \cdots +a_{1}(N) \\ &=2^{N-1}a_{1}(1)+2^{N-2}a_{1}(2)+\cdots +a_{1}(N) \\ &=2^{N-1}+2^{N-2}+\cdots +1 \\ &=\sum_{i_{1}=0}^{N-1}2^{i_{1}}. \end{aligned}$$
(2.25)
For \(k=3\) in (2.19), we have
$$\begin{aligned} a_{3}(N+1) &=3a_{3}(N)+a_{2}(N) \\ &=3 \bigl(3a_{3}(N-1)+a_{2}(N-1) \bigr)+a_{2}(N) \\ &=3^{2}a_{3}(N-1)+3a_{2}(N-1)+a_{2}(N) \\ &=\cdots \\ &=3^{N-2}a_{3}(3)+3^{N-3}a_{2}(3)+\cdots +a_{2}(N) \\ &=3^{N-2}a_{2}(2)+3^{N-3}a_{2}(3)+\cdots +a_{2}(N) \\ &=\sum_{i_{2}=0}^{N-2}3^{i_{2}}a_{2}(N-i_{2}). \end{aligned}$$
(2.26)
Then by (2.25) and (2.26), we get
$$\begin{aligned} a_{3}(N+1) &=\sum_{i_{2}=0}^{N-2}3^{i_{2}}a_{2}(N-i_{2}) \\ &=\sum_{i_{2}=0}^{N-2}3^{i_{2}}\sum _{i_{1}=0}^{N-2-i_{2}}2^{i_{1}} \\ &=\sum_{i_{2}=0}^{N-2}\sum _{i_{1}=0}^{N-2-i_{2}}3^{i_{2}}2^{i_{1}}. \end{aligned}$$
(2.27)
For \(k=4\) in (2.19), we have
$$\begin{aligned} a_{4}(N+1)&=4a_{4}(N)+a_{3}(N) \\ &=4 \bigl(4a_{4}(N-1)+a_{3}(N-1) \bigr)+a_{3}(N) \\ &=4^{2}a_{4}(N-1)+4a_{3}(N-1)+a_{3}(N) \\ &=\cdots \\ &=4^{N-3}a_{4}(4)+4^{N-4}a_{3}(4)+\cdots +a_{3}(N) \\ &=4^{N-3}a_{3}(3)+4^{N-4}a_{3}(4)+\cdots +a_{3}(N) \\ &=\sum_{i_{3}=0}^{N-3}4^{i_{3}}a_{3}(N-i_{3}). \end{aligned}$$
(2.28)
By (2.27) and (2.28), we have
$$\begin{aligned} a_{4}(N+1) &=\sum_{i_{3}=0}^{N-3}4^{i_{3}}a_{3}(N-i_{3}) \\ &=\sum_{i_{3}=0}^{N-3}4^{i_{3}}\sum _{i_{2}=0}^{N-3-i_{3}}\sum_{i_{1}=0} ^{N-3-i_{3}-i_{2}}3^{i_{2}}2^{i_{1}} \\ &=\sum_{i_{3}=0}^{N-3}\sum _{i_{2}=0}^{N-3-i_{3}}\sum_{i_{1}=0}^{N-3-i _{3}-i_{2}}4^{i_{3}}3^{i_{2}}2^{i_{1}}. \end{aligned}$$
(2.29)
Continuing this process, for \(2\leq k \leq N\), we have
$$ a_{k}(N+1)=\sum_{i_{k-1}=0}^{N-k+1}\sum _{i_{k-2}=0}^{N-k+1-i_{k-1}} \cdots \sum _{i_{1}=0}^{N-k+1-i_{k-1}-\cdots -i_{2}}k^{i_{k-1}}\cdots 2^{i_{1}}. $$
(2.30)

Therefore, we obtain the following differential equations.

Theorem 2.1

Let \(N \in \mathbb{N}\). Then the differential equations
$$ (N-1)!e^{-NF}=(-1)^{N-1}\sum_{k=1}^{N} \lambda^{N-k}(1+\lambda t)^{k} a _{k}(N)F^{(k)} $$
have a solution \(F=F(t)=\log (1+\frac{1}{\lambda }\log (1+\lambda t) )\), where
$$ a_{N}(N)=1,\qquad a_{1}(N)=1 $$
and
$$ a_{k}(N)=\sum_{i_{k-1}=0}^{N-k}\sum _{i_{k-2}=0}^{N-k-i_{k-1}}\cdots \sum _{i_{1}=0}^{N-k-i_{k-1}-\cdots -i_{2}}k^{i_{k-1}}\cdots 2^{i_{1}}. $$
From (2.1), we easily get
$$\begin{aligned} F &=\log \biggl(1+\frac{1}{\lambda }\log (1+\lambda t) \biggr) \\ &=\frac{\lambda \log (1+\frac{1}{\lambda }\log (1+\lambda t) )}{\log (1+\lambda t)}\cdot \frac{\log (1+\lambda t)}{\lambda } \\ &= \Biggl(\sum_{l_{1}=0}^{\infty }D_{l_{1},\lambda } \frac{t^{l_{1}}}{l _{1}!} \Biggr) \Biggl(\frac{1}{\lambda }\sum _{l_{2}=1}^{\infty }\frac{(-1)^{l _{2}-1}\lambda^{l_{2}}}{l_{2}}t^{l_{2}} \Biggr) \\ &= \Biggl(\sum_{l_{1}=0}^{\infty }D_{l_{1},\lambda } \frac{t^{l_{1}}}{l _{1}!} \Biggr) \Biggl(\sum_{l_{2}=1}^{\infty } \frac{(-\lambda )^{l_{2}-1}}{l _{2}}t^{l_{2}} \Biggr) \\ &=\sum_{l_{3}=1}^{\infty } \Biggl(\sum _{l_{1}=0}^{l_{3}-1}\frac{D_{l _{1},\lambda }}{l_{1}!}\cdot \frac{(-\lambda )^{l_{3}-l_{1}-1}}{(l _{3}-l_{1})} \Biggr)t^{l_{3}}. \end{aligned}$$
(2.31)
From (2.31), we get
$$\begin{aligned} F^{(k)} &= \biggl(\frac{d}{dt} \biggr)^{k} \Biggl\{ \sum _{l_{3}=1}^{\infty } \Biggl(\sum _{l_{1}=0}^{l_{3}-1}\frac{D_{l_{1},\lambda }}{l_{1}!}\cdot \frac{(-\lambda )^{l_{3}-l_{1}-1}}{(l_{3}-l_{1})} \Biggr)t^{l_{3}} \Biggr\} \\ &=\sum_{l_{3}=k}^{\infty } \Biggl(\sum _{l_{1}=0}^{l_{3}-1}\frac{D_{l _{1},\lambda }}{l_{1}!}\cdot \frac{(-\lambda )^{l_{3}-l_{1}-1}}{(l _{3}-l_{1})} \Biggr) (l_{3})_{k}t^{l_{3}-k} \\ &=\sum_{l_{3}=0}^{\infty } \Biggl(\sum _{l_{1}=0}^{l_{3}+k-1}\frac{D _{l_{1},\lambda }}{l_{1}!}\cdot \frac{(-\lambda )^{l_{3}+k-l_{1}-1}}{(l _{3}+k-l_{1})} \Biggr) (l_{3}+k)_{k}t^{l_{3}}. \end{aligned}$$
(2.32)
From (2.32), we get
$$\begin{aligned} (1+\lambda t)^{k} F^{(k)} &= \Biggl(\sum _{l=0}^{\infty }{k \choose l} \lambda^{l}t^{l} \Biggr) \Biggl\{ \sum _{l_{3}=0}^{\infty } \Biggl(\sum_{l _{1}=0}^{l_{3}+k-1} \frac{D_{l_{1},\lambda }}{l_{1}!}\cdot \frac{(- \lambda )^{l_{3}+k-l_{1}-1}}{(l_{3}+k-l_{1})} \Biggr) (l_{3}+k)_{k}t^{l_{3}} \Biggr\} \\ &= \Biggl(\sum_{l=0}^{\infty }(k)_{l} \lambda^{l}\frac{t^{l}}{l!} \Biggr) \Biggl\{ \sum _{l_{3}=0}^{\infty } \Biggl(\sum_{l_{1}=0}^{l_{3}+k-1} \frac{D _{l_{1},\lambda }}{l_{1}!}\cdot \frac{(-\lambda )^{l_{3}+k-l_{1}-1}}{(l _{3}+k-l_{1})} \Biggr) (l_{3}+k)!\frac{t^{l_{3}}}{l_{3}!} \Biggr\} \\ &=\sum_{n=0}^{\infty } \Biggl(\sum _{l_{3}=0}^{n}\sum_{l_{1}=0}^{l_{3}+k-1} \frac{(l _{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l_{1})}D_{l _{1},\lambda } \\ &\quad {}\times (-1)^{l_{3}+k-l_{1}-1}\lambda^{n+k-l_{1}-1} \Biggr) \frac{t^{n}}{n!} \end{aligned}$$
(2.33)
and also
$$\begin{aligned} e^{-NF} &=\sum_{m_{1}=0}^{\infty } \frac{(-N)^{m_{1}}}{m_{1}!}F^{m_{1}} \\ &=\sum_{m_{1}=0}^{\infty }\frac{(-N)^{m_{1}}}{m_{1}!} \biggl( \log \biggl(1+\frac{1}{\lambda }\log (1+\lambda t) \biggr) \biggr)^{m_{1}} \\ &=\sum_{m_{1}=0}^{\infty }(-N)^{m_{1}}\sum _{m_{2}=m_{1}}^{\infty }S _{1}(m_{2},m_{1}) \frac{1}{m_{2}!} \biggl(\frac{1}{\lambda }\log (1+ \lambda t) \biggr)^{m_{2}} \\ &=\sum_{m_{2}=0}^{\infty } \Biggl(\sum _{m_{1}=0}^{m_{2}}(-N)^{m_{1}}S _{1}(m_{2},m_{1}) \lambda^{-m_{2}} \Biggr)\frac{1}{m_{2}!} \bigl(\log (1+ \lambda t) \bigr)^{m_{2}} \\ &=\sum_{m_{2}=0}^{\infty } \Biggl(\sum _{m_{1}=0}^{m_{2}}(-N)^{m_{1}}S _{1}(m_{2},m_{1}) \lambda^{-m_{2}} \Biggr) \Biggl(\sum_{n=m_{2}}^{\infty }S _{1}(n,m_{2})\frac{\lambda^{n}t^{n}}{n!} \Biggr) \\ &=\sum_{n=0}^{\infty } \Biggl(\sum _{m_{2}=0}^{n}\sum_{m_{1}=0}^{m_{2}}(-N)^{m _{1}}S_{1}(m_{2},m_{1})S_{1}(n,m_{2}) \lambda^{n-m_{2}} \Biggr)\frac{t ^{n}}{n!}. \end{aligned}$$
(2.34)
Here \(S_{1}(n,k)\) is the Stirling number of the first kind.
Thus, by (2.13) and (2.33), we get
$$\begin{aligned} &(N-1)!e^{-NF} \\ &\quad =(-1)^{N-1}\sum_{k=1}^{N}\lambda^{N-k}a_{k}(N) \\ &\quad \quad {}\times\sum_{n=0} ^{\infty } \Biggl(\sum_{l_{3}=0}^{n}\sum _{l_{1}=0}^{l_{3}+k-1} \frac{(l_{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l _{1})}D_{l_{1},\lambda }(-1)^{l_{3}+k-l_{1}-1} \lambda^{n+k-l_{1}-1} \Biggr)\frac{t^{n}}{n!} \\ &\quad =\sum_{n=0}^{\infty } \Biggl((-1)^{N-1} \sum_{k=1}^{N}\lambda^{N-k}a _{k}(N)\sum_{l_{3}=0}^{n}\sum_{l_{1}=0}^{l_{3}+k-1} \frac{(l_{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l_{1})} \end{aligned}$$
(2.35)
$$\begin{aligned} &\quad \quad {}\times D_{l_{1},\lambda }(-1)^{l_{3}+k-l_{1}-1}\lambda^{n+k-l_{1}-1} \Biggr)\frac{t^{n}}{n!} \\ &\quad =\sum_{n=0}^{\infty } \Biggl(\sum _{k=1}^{N}\sum_{l_{3}=0}^{n} \sum_{l_{1}=0}^{l_{3}+k-1}(-1)^{N+l_{3}+k-l_{1}} \lambda^{N+n-l_{1}-1}a _{k}(N) \\ &\quad \quad {}\times \frac{(l_{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l _{1})}D_{l_{1},\lambda } \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.36)
By (2.34) and (2.36), we get
$$\begin{aligned} &(N-1)! \sum _{n=0}^{\infty } \Biggl(\sum_{m_{2}=0}^{n} \sum_{m_{1}=0}^{m _{2}}(-N)^{m_{1}}S_{1}(m_{2},m_{1})S_{1}(n,m_{2}) \lambda^{n-m_{2}} \Biggr)\frac{t^{n}}{n!} \\ &\quad =\sum_{n=0}^{\infty } \Biggl(\sum _{k=1}^{N}\sum_{l_{3}=0}^{n} \sum_{l_{1}=0}^{l_{3}+k-1}(-1)^{N+l_{3}+k-l_{1}} \lambda^{N+n-l_{1}-1}a _{k}(N) \frac{(l_{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l _{1})}D_{l_{1},\lambda } \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.37)

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
$$\begin{aligned} &(N-1)! \sum_{m_{2}=0}^{n}\sum _{m_{1}=0}^{m_{2}}(-N)^{m_{1}}S_{1}(m _{2},m_{1})S_{1}(n,m_{2}) \lambda^{n-m_{2}} \\ &\quad =\sum_{k=1}^{N}\sum _{l_{3}=0}^{n}\sum_{l_{1}=0}^{l_{3}+k-1}(-1)^{N+l _{3}+k-l_{1}} \lambda^{N+n-l_{1}-1}a_{k}(N) \\ &\quad \quad {}\times \frac{(l_{3}+k)!{n \choose l_{3}}(k)_{n-l_{3}}}{l_{1}!(l_{3}+k-l _{1})}D_{l_{1},\lambda }. \end{aligned}$$

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.

Declarations

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.

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 Mathematics, Kwangwoon University
(2)
Department of Mathematics Education and RINS, Gyeongsang National University
(3)
Division of Mathematics and Informational Statistics, Nanoscale Science and Technology Institute, Wonkwang University

References

  1. Kim, T, Kim, DS: Some identities of degenerate Daehee numbers arising from certain differential equations. J. Nonlinear Sci. Appl. 10, 744-751 (2017) MathSciNetView ArticleGoogle Scholar
  2. El-Desouky, B, Mustafa, A: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016) MathSciNetView ArticleGoogle Scholar
  3. Jang, G-W, Kim, T: Revisit of identities for Daehee numbers arising from nonlinear differential equations. Proc. Jangjeon Math. Soc. 20(2), 163-177 (2017) Google Scholar
  4. Kim, BM, Yun, SJ, Park, J-W: On a degenerate λ-q-Daeehee polynomials. J. Nonlinear Sci. Appl. 9(6), 4607-4616 (2016) MathSciNetMATHGoogle Scholar
  5. Kim, DS, Kim, T: Identities arising from higher-order Daehee polynomial bases. Open Math. 13, 196-208 (2015) MathSciNetMATHGoogle Scholar
  6. Kim, DS, Kim, T: Some identities for Bernoulli numbers of the second kind arising from a nonlinear differential equation. Bull. Korean Math. Soc. 52, 2001-2010 (2015) MathSciNetView ArticleMATHGoogle Scholar
  7. Kim, DS, Kim, T, Lee, S-H, Seo, J-J: Higher-order Daehee numbers and polynomials. Int. J. Math. Anal. 8(5-8), 273-283 (2014) MathSciNetView ArticleGoogle Scholar
  8. Bayad, A, Kim, T: Higher recurrences for Apostol-Bernoulli numbers. Russ. J. Math. Phys. 19(1), 1-10 (2012) MathSciNetView ArticleMATHGoogle Scholar
  9. Bayad, A, Kim, T: Identities for Apostol-type Frobenius-Euler polynomials resulting from the study of a nonlinear operator. Russ. J. Math. Phys. 23(2), 164-171 (2016) MathSciNetView ArticleMATHGoogle Scholar
  10. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equation. J. Number Theory 132, 2854-2865 (2012) MathSciNetView ArticleMATHGoogle Scholar
  11. Kim, T, Dolgy, DV, Kim, DS, Seo, JJ: Differential equations for Changhee polynomials and their applications. J. Nonlinear Sci. Appl. 9, 2857-2864 (2016) MathSciNetMATHGoogle Scholar
  12. Kim, T, Kim, DS: A note on nonlinear Changhee differential equations. Russ. J. Math. Phys. 23, 88-92 (2016) MathSciNetView ArticleMATHGoogle Scholar
  13. Kim, T, Kim, DS, Jang, LC, Kwon, HI: Differential equations associated with Mittag-Leffler polynomials. Glob. J. Pure Appl. Math. 12(4), 2839-2847 (2016) Google Scholar
  14. Kim, T, Kim, DS, Seo, JJ: Differential equations associated with degenerate Bell polynomials. Int. J. Pure Appl. Math. 108(3), 551-559 (2016) Google Scholar
  15. Kim, T, Kim, DS, Seo, J-J, Kwon, HI: Differential equations associated with λ-Changhee polynomials. J. Nonlinear Sci. Appl. 9, 3098-3111 (2016) MathSciNetMATHGoogle Scholar
  16. Kim, T, Seo, J-J: Revisit nonlinear differential equations arising from the generating functions of degenerate Bernoulli numbers. Adv. Stud. Contemp. Math. (Kyungshang) 26(3), 401-406 (2016) MATHGoogle Scholar
  17. Kwon, HI, Kim, T, Seo, J-J: A note on Daehee numbers arising from differential equations. Glob. J. Pure Appl. Math. 12(3), 2349-2354 (2016) Google Scholar
  18. Kwon, JK, Choi, YJ, Jang, MS, Yang, SO, Seong, MS: Some identities involving Changhee polynomials arising from a differential equations. Glob. J. Pure Appl. Math. 12(6), 4857-4866 (2016) Google Scholar
  19. Rim, SH, Jeong, JH, Park, J-W: Some identities involving Euler polynomials arising from a non-linear differential equation. Kyungpook Math. J. 53, 553-563 (2013) MathSciNetView ArticleMATHGoogle Scholar
  20. Yardimci, A, Simsek, Y: Identities for Korobov-type polynomials arising from functional equations and p-adic integral. J. Nonlinear Sci. Appl. 10, 2767-2777 (2017) View ArticleGoogle Scholar

Copyright

© The Author(s) 2017