Fully Degenerating of Daehee Numbers and Polynomials

: In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of Riemann integrals on the interval [ 0,1 ] . Finally, we derive their summation formulae.


Introduction
The generalizations of special polynomials have been one of the most emerging research fields in mathematical analysis and extensively investigated in order to find interesting identities and relations. Applications of the generalized special polynomials arise in problems of number theory, combinatorics, mathematical physics and other subareas of pure and applied mathematics provide motivation for introducing a new class of generalized polynomials. For example, some special polynomials occur in probability as the Edgeworth series; in combinatorics, they arise in the umbral calculus as an example of an Appell sequence which plays an important role in various problems connected with functional equations, interpolation problems, approximation theory, summation methods; in numerical analysis, they play a role in Gaussian quadrature; and in physics, they appear in quantum mechanical and optical beam transport problems. (see [1] for detail).
It is well known from [2] that where B (α) ,λ (ζ) are called λ-analogue of Bernoulli polynomials of higher order given via the following generating function: In [2], the degenerate Bernoulli polynomials of the second kind are defined by Note that lim The Daehee polynomials are known as, (see [7][8][9]).
In the case when ζ = 0, D j = D j (0) are called the Daehee numbers. The Equation (10) will be our main focus to proceed its fully degenerate version with their identities and properties with Section 2. In Section 3, we consider multiple fully degenerate Daehee polynomials of higher order which can be represented in terms of Riemann integrals on the interval [0, 1]. We derive their identities and properties among some other polynomials which will be mentioned in the next sections.

Fully Degenerating Daehee Numbers and Polynomials
Recall from Equation (6) that In the case λ approaches to 0, we see that the Equation (11) turns out to be classical one as follows: Note that e λ (log λ (ω)) = log λ (e λ (ω)) = ω. By making use of Equation (11), Kim et al. [5] introduced the new type of degenerate Daehee polynomials as follows: At the value ζ = 0, D j,λ = D j,λ (0) are called the degenerate Daehee numbers. Motivated by (12) and (13), we give the following definition. Definition 1. Let λ ∈ R−{0}. The fully degenerating Daehee polynomials are defined by means of the following generating function: Then, from (13), we see that Note that, lim λ→0dj,λ (ζ) =d j (ζ), (j ≥ 0). We note that ζ = 0,d j,λ :=d j,λ (0) are called the new type of fully degenerate Daehee numbers. The following identity will be useful for proving next theorem already known in [2]: Theorem 1. Let j ≥ 0, the following identity holds true: Proof. It is proved by using (11) that By comparing the coefficients of ω j j! on the above, we get the proof of this theorem.
Proof. By using (11), we get Thus, we get the result.
Proof. By using (11), we note that On the other hand, we have Thus, by (23) and (24), we arrive at the required the proof.

New Type of Higher-Order Fully Degenerating Daehee Numbers and Polynomials
Let us define the new type of fully degenerate Daehee polynomials of order r (∈ N) by the following multiple Riemann integral on the interval [0, 1] In the case when ζ = 0,d j,λ =d (r) j,λ (0) are called the new type of fully degenerate Daehee numbers of the order r.
Proof. From (25), we note that Thus, by comparing the coefficients of ω j j! on the above, we obtain the result.
Proof. Using (25), we have Therefore, by comparing the coefficients of ω j j! on the above, we arrive at the desired result.
Proof. By replacing ω by e λ (ω) − 1 in (25), we get On the other hand, we have By (26) and (27), we complete the proof.
From (2), we get Proof. By changing t to log λ (1 + t) in (28), we get In view of (25) and (29), we get the result.
Proof. From (25), we note that In view of (34), we complete the proof.
Proof. From 25, we note that Therefore, by (25) and (36), we complete the proof.
Theorem 15. For r, k ∈ N, with r > k, we havẽ we have Therefore, by (25) and (37), we obtain the result.
Theorem 16. For j ≥ 0, we havẽ Proof. Now, we observe that Equating the coefficients of ω j j! on both sides of the above, we get the result.