Degenerate Daehee Numbers of the Third Kind

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.


Introduction
After Carlitz [1,2], many mathematicians have studied degenerate functions and numbers (see [3,9,15,16,17,18,19,23,24]).They mainly used (1+λt) 1 λ instead of e t to degenerate polynomials and numbers.In [17], T. Kim and D.S. Kim called (1+ λt) 1 λ the degenerate exponential function and express it as e t λ .They also presented the degenerate gamma function and degenerate Laplace transformation using e t λ .In [25], authors introduced four kind degenerate version of Cauchy numbers.In the degenerate Cauchy numbers of the first and second kind, (1 + λt) 1 λ was used instead of e t , We call this degenerate based on the exponential sense.
In accordance with exponential sense, log(1 + λt) λ can be used for t to study degenerate numbers and polynomials.In this sense, in [15], T. Kim introduced the degenerate Cauchy numbers by the generating function to be log e λ (t) − 1 log(1 + log e λ (t)) = log(1 + λt) It is natural to think a degenerate log function as the inverse function of the degenerate exponential function.The degenerate log function, denoted by log λ (t), is defined by the generating function to be log λ (t) = 1 λ (t λ − 1) (see [19]). ( The Cauchy numbers or the second kind Bernoulli numbers, denoted by C n , are defined by the generating function to be [15,18,21,25]).
Recently, in [18], T. Kim introduced the degenerate Cauchy numbers by the generating function to be If λ goes to 0, then C n (λ) converges to C n .In this case, author used log λ (t) instead of log t 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 : The Bernoulli numbers, which started with a study on the sum of the power series, has many relationships with other special numbers [2,4,5,3,6,7,13,21,22,27].
Although the Daehee numbers are readily available as D n = (−1) n n! n+1 , the Daehee numbers have many interesting relationships with other special numbers.For example, the following show relationships between the Daehee numbers and the Bernoulli numbers.
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 [11]).
The degenerate Daehee numbers, denoted by D λ (n), are introduced as follows(see [9]): If λ goes to 0, then D λ (n) converges to D n .This degenerate Daehee numbers D λ (n) are degenerate based on the exponential sense.Recently, D. S. Kim and et al. presented the degenerate Daehee polynomials and numbers of the second kind as follows [12].log(1 + t) degenerate Daehee numbers of the 2nd kind degenerate Daehee numbers of the 3rd kind Table 1.Three kinds of degenerate Daehee numbers 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 λ,3 (n), as follows.
The Table 1 summarizes the three types of degenerate Daehee numbers.
In this paper, we define the degenerate Daehee numbers based on the log sense.And 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.

Degenerate Daehee numbers of the third kind
From now on, for any real x and non negative integer n, we denote (x) n for falling factorial (x ) and (x) 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 This ( 12) yields the following.
From (13), it is easy to show that lim λ→0 D λ,3 (n) = D n .Now let us investigate the relationship between the Daehee numbers and the degenerate Daehee numbers.
The equation ( 14) yields a relationship between the Daehee numbers and the degenerate Daehee numbers.

Theorem 1. For any nonnegative integer n,
From the definition of the Daehee numbers ( 6) and the degenerate Daehee numbers (10), we get the following.
From the equation ( 15), we have a kind of inversion formula for Theorem 1.
Theorem 2. For any nonnegative integer n, In [15], the degenerate Stirling numbers of the first kind, denoted by S 1,λ (n, k), are introduced by The equation (7) notice that the Daehee numbers can be represented by the Bernoulli and Stirling numbers of the first kind.The next equation is a degenerate version of the equation (7). )) By comparing the coefficients in the equation ( 17), we can obtain the following theorem.

Theorem 3. For any nonnegative integer n,
In [15], the degenerate Stirling numbers of the second kind were defined by the generating function Using (18) and the definition of the degenerate Bernoulli numbers and the degenerate Daehee numbers, we get the following.
The equation (19) give us an inversion formula of Theorem 3 which is a degenerate version of the equation (8).
Obtain the following theorem from the definition of the degenerate Daehee numbers of the third kind (10) and from the previous (20).
Theorem 7.For any nonnegative integer n,

Differential equations arising from the generating function of degenerate Daehee numbers
From now on, we use F = F (t) to denote the degenerate log function : and for a natural number N , F (N ) to denote the N -th derivative of F , that is, 1) .
Differentiating the two sides of the equation ( 25) results in the following: From the observation ( 26) and ( 27), we assume that Let us take differentiate both sides of the ( 28), then we have (29) From ( 28) and (29), mathematical induction gives us the following theorem.Theorem 8.For any positive integer N , the differential equation We note that On the other hand

Conclusion
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 this view points, 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.