Degenerate Daehee Numbers of the Third Kind

Abstract

In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

1. Introduction

After Carlitz [1,2], many mathematicians have studied degenerate functions and numbers (see [3,4,5,6,7,8,9,10,11]). They mainly used ${(1+\lambda t)}^{\frac{1}{\lambda }}$ instead of $e^t$ to degenerate polynomials and numbers. In [7], T. Kim and D.S. Kim called ${(1+\lambda t)}^{\frac{1}{\lambda }}$ the degenerate exponential function and expressed it as $e_{\lambda }^t$. They also presented the degenerate gamma function and degenerate Laplace transformation using $e_{\lambda }^t$. In [12], the authors introduced four degenerate versions of Cauchy numbers. In the degenerate Cauchy numbers of the first and second kind, ${(1+\lambda t)}^{\frac{1}{\lambda }}$ was used instead of $e^t$. We call this degenerate based on the exponential sense.

In accordance with the exponential sense, $log{(1+\lambda t)}^{\frac{1}{\lambda }}$ can be used for t to study degenerate numbers and polynomials. It is natural to think of a degenerate log function as the inverse function of the degenerate exponential function. The degenerate log function, denoted by ${log}_{\lambda }\left(t\right)$, is defined by the generating function to be

$${log}_{\lambda }\left(t\right)=\frac{1}{\lambda }(t^{\lambda }-1)$$

(1)

(see [9]).

The Cauchy numbers or the second kind Bernoulli numbers, denoted by $C_n$, are introduced in [5,8,12,13]. Recently, in [8], T. Kim introduced the degenerate Cauchy numbers.

In this case, the author used ${log}_{\lambda }\left(t\right)$ instead of $logt$ for degenerating. We call this degenerate based on the log sense.

As is well known, the Bernoulli numbers, denoted by $B_n$, are defined by the generating function:

$$\frac{t}{e^t-1}=\sum _{n=0}^{\infty }{B}_n\frac{t^n}{n!}.$$

(2)

The Bernoulli numbers, which started with a study on the sum of the power series, has many relationships with other special numbers [2,3,13,14,15,16,17,18,19,20,21]. In [1], the degenerate Bernoulli numbers are presented as follows:

$$\frac{t}{e_{\lambda }\left(t\right)-1}=\frac{t}{{(1+\lambda t)}^{\frac{1}{\lambda }}-1}=\sum _{n=0}^{\infty }{\beta }_{n,\lambda }\frac{t^n}{n!}.$$

(3)

The Daehee numbers, denoted by $D_n$, are defined by the generating function

$$\frac{log(1+t)}{t}=\sum _{n=0}^{\infty }{D}_n\frac{t^n}{n!},$$

(4)

(see [4,10,11,14,15,20,22]).

The degenerate Daehee numbers $D_{\lambda }\left(n\right)$, are introduced as $\frac{\lambda log(1+\frac{1}{\lambda }log(1+\lambda t))}{log(1+\lambda t)}={\sum }_{n=0}^{\infty }{D}_{\lambda }\left(n\right)\frac{t^n}{n!}$ (see [4]). If $\lambda$ goes to 0, then $D_{\lambda }\left(n\right)$ converges to $D_n$. These degenerate Daehee numbers $D_{\lambda }\left(n\right)$ are degenerate based on the exponential sense. Recently, D.S. Kim et al. presented degenerate Daehee polynomials and numbers of the second kind as follows [23].

$$\frac{log(1+t)}{{(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1}{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{D}_{\lambda ,2}(n,x)\frac{t^n}{n!}.$$

(5)

When $x=0,D_{\lambda ,2}\left(n\right)=D_{\lambda ,2}(n,0)$ are called the degenerate Daehee numbers of the second kind. These degenerate numbers are also based on the exponential sense.

It is natural to think about degenerate Daehee numbers based on the log sense. We define the degenerate Daehee numbers of the third kind, denoted by $D_{\lambda ,3}\left(n\right)$, as follows.

$$\frac{{log}_{\lambda }(1+t)}{t}=\frac{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}{t}=\sum _{n=0}^{\infty }{D}_{\lambda ,3}\left(n\right)\frac{t^n}{n!}.$$

(6)

We note that ${lim}_{\lambda \to 0}{D}_{\lambda ,3}\left(n\right)=D_n$ for each n.

In this paper, we define the degenerate Daehee numbers based on the log sense. We obtain some identities which are connected with the Daehee, the degenerate Daehee, and the degenerate Daehee numbers of the second kind. Additionally, we deduce a differential equation using the degenerate log function, and we derive some identities related to the degenerate Daehee numbers from this differential equation.

2. Degenerate Daehee Numbers of the Third Kind

From now on, for any real x and non-negative integer n, we denote ${\left(x\right)}_n$ for falling factorial ${\left(x\right)}_n=x(x-1)(x-2)\cdots (x-n+1)$ and ${\left(x\right)}_0=1$. We use $S_1(m,n)$ and $S_2(m,n)$ to denote the Stirling number of the first kind and the second kind, respectively. From the definition of the degenerate Daehee numbers of the third kind, we get

$$\begin{array}{cc}\hfill \frac{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}{t}& =\frac{1}{\lambda t}\sum _{n=1}^{\infty }{\left(\lambda \right)}_n\frac{t^n}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\frac{{\left(\lambda \right)}_{n+1}}{\lambda (n+1)}\frac{t^n}{n!}\hfill \end{array}.$$

(7)

Equation (7) yields the following.

$$D_{\lambda ,3}\left(n\right)=\frac{{\left(\lambda \right)}_{n+1}}{\lambda (n+1)}=\frac{\lambda (\lambda -1)(\lambda -2)\cdots (\lambda -n)}{\lambda (n+1)}.$$

(8)

From Equation (8), it is easy to show that ${lim}_{\lambda \to 0}{D}_{\lambda ,3}\left(n\right)=D_n$.

Now let us investigate the relationship between the Daehee numbers and the degenerate Daehee numbers.

$$\begin{array}{cc}\hfill \frac{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}{t}& =\frac{log\left(e^{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}-1+1\right)}{t}=\sum _{l=0}^{\infty }{D}_l\frac{{\left(e^{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}-1\right)}^l}{l!}\hfill \\ & =\sum _{l=0}^{\infty }{D}_l\sum _{m=l}^{\infty }{S}_2(m,l)\frac{1}{m!}\frac{1}{{\lambda }^m}{\left({(1+t)}^{\lambda }-1\right)}^m\hfill \\ & =\sum _{l=0}^{\infty }{D}_l\sum _{m=l}^{\infty }{S}_2(m,l)\frac{1}{m!}\frac{1}{{\lambda }^m}\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{m}{k}\right){(-1)}^{m-k}\sum _{n=0}^{\infty }{\left(k\lambda \right)}_n\frac{t^n}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{l=0}^m\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{m}{k}\right){(-1)}^{m-k}\frac{D_l{S}_2(m,l){\left(k\lambda \right)}_n}{m!{\lambda }^m}\frac{t^n}{n!}\hfill \end{array}.$$

(9)

Equation (9) yields a relationship between the Daehee numbers and the degenerate Daehee numbers.

Theorem 1.

For any nonnegative integer n,

$$D_{\lambda ,3}\left(n\right)=\sum _{m=0}^{\infty }\sum _{l=0}^m\sum _{k=0}^m\left(\genfrac{}{}{0pt}{}{m}{k}\right){(-1)}^{m-k}\frac{D_l{S}_2(m,l){\left(k\lambda \right)}_n}{m!{\lambda }^m}.$$

From the definition of the Daehee numbers, Equation (4), and the degenerate Daehee numbers, Equation (6), we get the following.

$$\begin{array}{cc}\hfill \frac{log(1+t)}{t}& =\frac{\frac{1}{\lambda }\left\{{\left(1+\left({(\lambda log(1+t)+1)}^{\frac{1}{\lambda }}-1\right)\right)}^{\lambda }-1\right\}}{t}\hfill \\ & =\sum _{l=0}^{\infty }{D}_{\lambda ,3}\left(l\right)\frac{{\left({(\lambda log(1+t)+1)}^{\frac{1}{\lambda }}-1\right)}^l}{l!}\hfill \\ & =\sum _{l=0}^{\infty }\frac{D_{\lambda ,3}\left(l\right)}{l!}\sum _{m=0}^l{(-1)}^{l-m}\left(\genfrac{}{}{0pt}{}{l}{m}\right){(\lambda log(1+t)+1)}^{\frac{m}{\lambda }}\hfill \\ & =\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\sum _{m=0}^l\sum _{k=0}^n{(-1)}^{l-m}\left(\genfrac{}{}{0pt}{}{l}{m}\right){\left(\frac{m}{\lambda }\right)}_k{\lambda }^k\frac{D_{\lambda ,3}\left(l\right)S_1(n,k)}{l!}\frac{t^n}{n!}\hfill \end{array}.$$

(10)

From Equation (10), we have a kind of inversion formula for Theorem 1.

Theorem 2.

For any nonnegative integer n,

$$D_n=\sum _{l=0}^{\infty }\sum _{m=0}^l\sum _{k=0}^n{(-1)}^{l-m}\left(\genfrac{}{}{0pt}{}{l}{m}\right){\left(\frac{m}{\lambda }\right)}_k{\lambda }^k\frac{D_{\lambda ,3}\left(l\right)S_1(n,k)}{l!}.$$

In [5], the degenerate Stirling numbers of the first kind, denoted by $S_{1,\lambda }(n,k)$, are introduced by

$$\frac{1}{k!}{\left({log}_{\lambda }(1+t)\right)}^k=\sum _{n=k}^{\infty }{S}_{1,\lambda }(n,k)\frac{t^n}{n!}.$$

(11)

The following show relationships between the Daehee numbers and the Bernoulli numbers.

$$\begin{array}{cc}\hfill D_n& =\sum _{m=0}^n{B}_m{S}_1(n,m)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill B_m& =\sum _{n=0}^m{D}_n{S}_2(m,n)\hfill \end{array}$$

(13)

where $S_1(n,m)$ and $S_2(n,m)$ are the the Stirling numbers of the first kind and the second kind, respectively (see [24]).

In Equation (12), the Daehee numbers can be represented by the Bernoulli and Stirling numbers of the first kind. The next equation is a degenerate version of Equation (12).

$$\begin{array}{cc}\hfill \frac{{log}_{\lambda }(1+t)}{t}& =\frac{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}{t}\hfill \\ & =\frac{\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)}{{\left(1+\lambda \left(\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)\right)\right)}^{\frac{1}{\lambda }}-1}\hfill \\ & =\sum _{l=0}^{\infty }{\beta }_{l,\lambda }\frac{{\left(\frac{1}{\lambda }\left({(1+t)}^{\lambda }-1\right)\right)}^l}{l!}\hfill \\ & =\sum _{n=0}^{\infty }\sum _{l=0}^n{\beta }_{l,\lambda }{S}_{1,\lambda }(n,l)\frac{t^n}{n!}\hfill \end{array}.$$

(14)

By comparing the coefficients in Equation (14), we can obtain the following theorem.

Theorem 3.

For any nonnegative integer n,

$$D_{\lambda ,3}\left(n\right)=\sum _{l=0}^n{\beta }_{l,\lambda }{S}_{1,\lambda }(n,l).$$

In [5], the degenerate Stirling numbers of the second kind were defined by the generating function

$$\frac{1}{k!}{\left(e_{\lambda }\left(t\right)-1\right)}^k=\frac{{\left({(1+\lambda t)}^{\frac{1}{\lambda }}-1\right)}^k}{k!}=\sum _{n=k}^{\infty }{S}_{2,\lambda }(n,k)\frac{t^n}{n!}.$$

(15)

Using Equation (15) and the definition of the degenerate Bernoulli numbers and the degenerate Daehee numbers, we get the following.

$$\begin{array}{cc}\hfill \frac{t}{{(1+\lambda t)}^{\frac{1}{\lambda }}-1}& =\frac{\frac{1}{\lambda }\left({\left(1+({(1+\lambda t)}^{\frac{1}{\lambda }}-1)\right)}^{\lambda }-1\right)}{{(1+\lambda t)}^{\frac{1}{\lambda }}-1}\hfill \\ & =\sum _{l=0}^{\infty }{D}_{\lambda ,3}\left(l\right)\frac{1}{l!}{\left({(1+\lambda t)}^{\frac{1}{\lambda }}-1\right)}^l\hfill \\ & =\sum _{n=l}^{\infty }\sum _{l=0}^n{D}_{\lambda ,3}\left(l\right)S_{2,\lambda }(n,l)\frac{t^n}{n!}\hfill \end{array}.$$

(16)

Equation (16) gives us an inversion formula of Theorem 3, which is a degenerate version of Equation (13).

Theorem 4.

For any nonnegative integer n,

$${\beta }_{n,\lambda }=\sum _{l=0}^n{D}_{\lambda ,3}\left(l\right)S_{2,\lambda }(n,l).$$

Now let us observe a relation between the degenerate Daehee numbers of the third kind and the Bernoulli numbers.

$$\begin{array}{cc}\hfill \frac{{log}_{\lambda }(1+t)}{t}& =\frac{{log}_{\lambda }(1+t)}{e^{{log}_{\lambda }(1+t)}-1}\frac{e^{{log}_{\lambda }(1+t)}-1}{t}\hfill \\ & =\sum _{l=0}^{\infty }{B}_l\frac{{\left({log}_{\lambda }(1+t)\right)}^l}{l!}\sum _{m=1}^{\infty }\frac{1}{t}\frac{{\left({log}_{\lambda }(1+t)\right)}^m}{m!}\hfill \\ & =\sum _{l=0}^{\infty }{B}_l\sum _{k=l}^{\infty }{S}_{1,\lambda }(k,l)\frac{t^k}{k!}\sum _{m=1}^{\infty }\sum _{n=m}^{\infty }{S}_{1,\lambda }(n,m)\frac{t^{n-1}}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\left(\sum _{k=0}^n\sum _{l=0}^{n-k}\sum _{m=1}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_l{S}_{1,\lambda }(n-k,l)S_{1,\lambda }(n+1,m)}{n+1}\right)\frac{t^n}{n!}\hfill \end{array}.$$

(17)

Obtain the following theorem from the definition of the degenerate Daehee numbers of the third kind (6) and from the previous Equation (17).

Theorem 5.

For any nonnegative integer n,

$$D_{\lambda ,3}\left(l\right)=\sum _{k=0}^n\sum _{l=0}^{n-k}\sum _{m=1}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_l{S}_{1,\lambda }(n-k,l)S_{1,\lambda }(n+1,m)}{n+1}.$$

Substituting $e^{{log}_{\lambda }(1+t)}-1$ by t in the definition of the second kind (5), the left side of Equation (5) becomes

$$\frac{log(1+\left(e^{{log}_{\lambda }(1+t)}-1\right))}{(1+\lambda log{(1+e^{{log}_{\lambda }(1+t)}-1))}^{\frac{1}{\lambda }}-1}=\frac{{log}_{\lambda }(1+t)}{t},$$

(18)

and the right side becomes

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{D}_{\lambda ,2}\left(k\right)\frac{{(e^{{log}_{\lambda }(1+t)}-1)}^k}{k!}& =\sum _{k=0}^{\infty }{D}_{\lambda ,2}\left(k\right)\sum _{l=k}^{\infty }{S}_{2,\lambda }(l,k)\frac{{\left({log}_{\lambda }(1+t)\right)}^l}{l!}\hfill \\ & =\sum _{k=0}^{\infty }{D}_{\lambda ,2}\left(k\right)\sum _{l=k}^{\infty }{S}_{2,\lambda }(l,k)\sum _{n=l}^{\infty }{S}_{1,\lambda }(n,l)\frac{t^n}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\sum _{l=0}^n\sum _{k=0}^l{D}_{\lambda ,2}\left(k\right)S_{2,\lambda }(l,k)S_{1,\lambda }(n,l)\frac{t^n}{n!}\hfill \end{array}.$$

(19)

The following theorem is obtained by comparing the coefficients of Equation (18) with Equation (19).

Theorem 6.

For any nonnegative integer n,

$$D_{\lambda ,3}\left(n\right)=\sum _{l=0}^n\sum _{k=0}^l{D}_{\lambda ,2}\left(k\right)S_{2,\lambda }(l,k)S_{1,\lambda }(n,l).$$

To represent $D_{\lambda ,3}\left(n\right)$ as $D_{\lambda ,2}\left(n\right)$, substituting $e_{\lambda }(log(1+t))-1={(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1$ by t in the definition of the degenerate Daehee numbers of the third kind, Equation (6), the left side becomes

$$\begin{array}{cc}\hfill \frac{{log}_{\lambda }(1+({(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1))}{{(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1}& =\frac{\frac{1}{\lambda }({(1+({(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1))}^{\lambda }-1)}{{(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1}\hfill \\ & =\frac{log(1+t)}{{(1+\lambda log(1+t))}^{\frac{1}{\lambda }}-1}\hfill \end{array},$$

(20)

and the right side becomes

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{D}_{\lambda ,3}\left(k\right)\frac{{(e_{\lambda }(log(1+t))-1)}^k}{k!}& =\sum _{k=0}^{\infty }{D}_{\lambda ,3}\left(k\right)\sum _{l=k}^{\infty }{S}_{2,\lambda }(l,k)\frac{{(log(1+t))}^l}{l!}\hfill \\ & =\sum _{k=0}^{\infty }{D}_{\lambda ,3}\left(k\right)\sum _{l=k}^{\infty }{S}_{2,\lambda }(l,k)\sum _{n=l}^{\infty }{S}_1(n,l)\frac{t^n}{n!}\hfill \\ & =\sum _{n=0}^{\infty }\sum _{l=0}^n\sum _{k=0}^l{D}_{\lambda ,3}\left(k\right)S_{2,\lambda }(l,k)S_1(n,l)\frac{t^n}{n!}\hfill \end{array}.$$

(21)

From Equations (20) and (21), we obtain an inversion identity of Theorem 6.

Theorem 7.

For any nonnegative integer n,

$$D_{\lambda ,2}\left(n\right)=\sum _{l=0}^n\sum _{k=0}^l{D}_{\lambda ,3}\left(k\right)S_{2,\lambda }(l,k)S_1(n,l).$$

3. Differential Equations Arising from the Generating Function of Degenerate Daehee Numbers

In this section, we use techniques similar to those of Martin for our results (see [25,26,27]). From now on, we use $F=F\left(t\right)$ to denote the degenerate log function:

$$F\left(t\right)={log}_{\lambda }(1+t)=\frac{1}{\lambda }({(1+t)}^{\lambda }-1)$$

(22)

and, for a natural number N, $F^{\left(N\right)}$ to denote the N-th derivative of F; that is,

$$F^{\left(0\right)}=F\left(t\right),F^{\left(N\right)}=\frac{d}{dt}{F}^{(N-1)}.$$

Differentiating the two sides of Equation (22) results in the following:

$$\begin{array}{cc}\hfill \frac{d}{dt}F(t)& =\frac{d}{dt}{log}_{\lambda }(1+t)={(1+t)}^{\lambda -1}\hfill \\ & =\frac{1}{1+t}\left\{\frac{\lambda }{\lambda }({(1+t)}^{\lambda }-1+1)\right\}\hfill \\ & =\frac{\lambda }{1+t}\frac{1}{\lambda }({(1+t)}^{\lambda }-1)+\frac{1}{1+t}\hfill \\ & =\frac{\lambda }{1+t}F(t)+\frac{1}{1+t}\hfill \end{array}.$$

(23)

Further differentiating Equation (23) yields

$$\begin{array}{cc}\hfill \frac{d^2}{d{t}^2}F\left(t\right)& =-\frac{\lambda }{{(1+t)}^2}F\left(t\right)+\frac{\lambda }{1+t}{F}^{\prime }\left(t\right)-\frac{1}{{(1+t)}^2}\hfill \\ & =-\frac{\lambda }{{(1+t)}^2}F\left(t\right)+\frac{\lambda }{1+t}\left\{\frac{\lambda }{1+t}F\left(t\right)+\frac{1}{1+t}\right\}-\frac{1}{{(1+t)}^2}\hfill \\ & =-\frac{\lambda (\lambda -1)}{{(1+t)}^2}F\left(t\right)++\frac{\lambda -1}{{(1+t)}^2}\hfill \end{array}.$$

(24)

Based on Equations (23) and (24), we assume that

$$F^{\left(N\right)}=\frac{{\left(\lambda \right)}_N}{{(1+t)}^N}F+\frac{{\left(\lambda \right)}_N}{\lambda {(1+t)}^N}.$$

(25)

Let us differentiate both sides of Equation (25). Then we have

$$\begin{array}{cc}\hfill F^{(N+1)}& =\frac{-N}{{(1+t)}^{N+1}}{\left(\lambda \right)}_{N}F+\frac{1}{{(1+t)}^N}{\left(\lambda \right)}_N{F}^{\left(1\right)}+\frac{-N}{\lambda {(1+t)}^{N+1}}{\left(\lambda \right)}_N\hfill \\ & =\frac{{\left(\lambda \right)}_N}{{(1+t)}^{N+1}}(\lambda -N)F+\frac{{\left(\lambda \right)}_N}{\lambda {(1+t)}^{N+1}}(\lambda -N)\hfill \\ & =\frac{{\left(\lambda \right)}_{N+1}}{{(1+t)}^{N+1}}F+\frac{{\left(\lambda \right)}_{N+1}}{\lambda {(1+t)}^{N+1}}\hfill \end{array}.$$

(26)

From Equations (25) and (26), a mathematical induction gives us the following theorem.

Theorem 8.

For any positive integer N, the differential equation

$$F^{\left(N\right)}=\frac{{\left(\lambda \right)}_N}{{(1+t)}^N}F+\frac{{\left(\lambda \right)}_N}{\lambda {(1+t)}^N}$$

(27)

has a solution

$$F\left(t\right)={log}_{\lambda }(1+t)=\frac{1}{\lambda }({(1+t)}^{\lambda }-1).$$

We note that

$$\begin{array}{cc}\hfill F^{\left(N\right)}& =\frac{d^N}{d{t}^N}{log}_{\lambda }(1+t)=\frac{d^N}{d{t}^N}\left(\frac{lo{g}_{\lambda }(1+t)}{t}·t\right)\hfill \\ & =\frac{d^N}{d{t}^N}\left(\sum _{n=1}^{\infty }n{D}_{\lambda ,3}(n-1)\frac{t^n}{n!}\right)\hfill \\ & =\sum _{n=0}^{\infty }(n+N)D_{\lambda ,3}(n+N-1)\frac{t^m}{m!}\hfill \end{array}.$$

(28)

On the other hand,

$$\begin{array}{cc}\hfill \frac{1}{{(1+t)}^N}F& =\sum _{l=0}^{\infty }\left(\genfrac{}{}{0pt}{}{N+l-1}{l}\right){(-1)}^l{t}^l\left({log}_{\lambda }(1+t)\right)\hfill \\ & =\sum _{l=0}^{\infty }\left(\genfrac{}{}{0pt}{}{N+l-1}{l}\right){(-1)}^l{t}^l\sum _{m=1}^{\infty }\frac{1}{\lambda }{\left(\lambda \right)}_m\frac{t^m}{m!}\hfill \\ & =\sum _{n=1}^{\infty }\sum _{m=1}^n\frac{1}{\lambda }{\left(\lambda \right)}_m\left(\genfrac{}{}{0pt}{}{N+n-m-1}{N-1}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)(n-m)!\frac{t^n}{n!}\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill \frac{{\left(\lambda \right)}_N}{\lambda }\frac{1}{{(1+t)}^N}& =\frac{{\left(\lambda \right)}_N}{\lambda }\sum _{n=0}^{\infty }{(-1)}^n\left(\genfrac{}{}{0pt}{}{N+n-1}{n}\right)t^n\hfill \\ & =\sum _{n=0}^{\infty }{(-1)}^n\frac{{\left(\lambda \right)}_N}{\lambda }n!\left(\genfrac{}{}{0pt}{}{N+n-1}{n}\right)\frac{t^n}{n!}\hfill \end{array}.$$

(30)

As a result of Equation (25), by comparing the coefficients of Equations (26), (27), and (28), we get the following identity.

Theorem 9.

For any nonnegative integer m and N,

$$\begin{array}{c}(n+N)D_{\lambda ,3}(n+N-1)\hfill \\ =\left\{\begin{array}{cc}\sum _{m=1}^n\frac{1}{\lambda }{\left(\lambda \right)}_m\left(\genfrac{}{}{0pt}{}{N+n-m-1}{N-1}\right)\left(\genfrac{}{}{0pt}{}{n}{m}\right)(n-m)!+{(-1)}^n\frac{{\left(\lambda \right)}_N}{\lambda }n!\hfill & ifn\ge 1\hfill \\ \frac{{\left(\lambda \right)}_N}{\lambda }\hfill & ifn=0\hfill \end{array}\right..\hfill \end{array}$$

In the other view point of the right side of Equation (25), we get

$$\begin{array}{cc}\hfill \frac{{\left(\lambda \right)}_N}{{(1+t)}^N}F+\frac{{\left(\lambda \right)}_N}{\lambda {(1+t)}^N}& =\frac{{(\lambda -1)}_{N-1}}{{(1+t)}^N}\left(\lambda F+1\right)\hfill \\ & =\frac{{(\lambda -1)}_{N-1}}{{(1+t)}^N}\left(\lambda \frac{1}{\lambda }({(1+t)}^{\lambda }-1)+1\right)\hfill \\ & ={(\lambda -1)}_{N-1}{(1+t)}^{\lambda -N}\hfill \\ & =\sum _{n=0}^{\infty }{(\lambda -1)}_{N-1}{(\lambda -N)}_n\frac{t^n}{n!}\hfill \end{array}.$$

(31)

Equations (26) and (29) yield the following.

Theorem 10.

For nonnegative integer n and N,

$$(n+N)D_{\lambda ,3}(n+N-1)={(\lambda -1)}_{N-1}{(\lambda -N)}_n.$$

4. Results and Discussion

In this paper, we have studied the degenerate Daehee numbers of the third kind. To define the degenerate function, we used degenerate log function ${log}_{\lambda }\left(t\right)=\frac{1}{\lambda }({(1+t)}^{\lambda }-1)$. First we defined the degnerate Daehee numbers of the third kind. We obtained two relations between the Daehee numbers and the degenerate Daehee numbers of the second kind with Stirling numbers in Theorem 1 and Theorem 2. After that, we have two relationships between the degenerate Daehee numbers of the third kind and the degenerate Bernoulli numbers, in Theorem 3 and Theorem 4. We obtained some relations between the degenerate Daehee numbers of the second kind and the third kind in Theorem 6 and Theorem 7.

In Section 3, we derived a differential equation from the degenerate log function, Theorem 6. From the differential equation in Theorem 6, we deduced some identities for the degenerate Daehee numbers of the third kind in Theorem 10.

5. Conclusions

In [1,2], L. Carlitz considered the degenerate exponential function. By using this degenerate exponential function, he studied the degenerate Bernoulli numbers and polynomials, which are given by the generating function. In the degenerate function, if we take $\lambda$ to 0 and take the appropriate variable t, we can get some form of polynomial or special polynomials number. From this view point, we consider the inverse function of Carlitz’s degenerate exponential function, which is called the degenerate logarithmic function. From our degenerate logarithmic function, we derive several identities of special numbers. Through our results, we are able to see that degenerate log function is a useful tool for study of special number theory. Finally we show that degenerate Daehee numbers of third kind are the solution of the non-linear differential equations in Equation (27).

Recently in [28], T. Kim and D.S. Kim introduced some relations of trigonometric numbers through mathematical physics of view points. By taking the appropriate value $\lambda$ and variable t in this our results, you can derive interesting results for special numbers useful in combinatorial and mathematical physics.