On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we deﬁne the fully degenerate Daehee polynomials of the second kind which are diﬀerent from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.


Introduction
Let p be a fixed prime number. roughout this paper, Z p , Q p , and C p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively.
In this paper, we define the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind and investigate identities and properties of new polynomials.

Fully Degenerate Daehee Polynomials of the Second Kind
Let us assume that λ ∈ R. By (3), we have By (24), we define the degenerate Daehee polynomials of the second kind by the generating function to be In the special case, x � 0, and D n (λ) � D n (0 | λ) are called the degenerate Daehee numbers of the second kind.
Note that and thus, we know that From (7) and (24), we have By (25) and (28), we have Since Journal of Mathematics by (29) and (30), we have us, by (29) and (30), we have the following theorem which is Witt's type formula about degenerate Daehee polynomials of the second kind.
By replacing t as e t − 1 in (25), we obtain the following: On the other hand, where B n (x | λ) is the degenerate Bernoulli polynomials of the second kind of order r ∈ Z which are defined by the generating function to be (see [20,25]).
In particular, if r � 1, is called the degenerate Bernoulli polynomials of the second kind.

Journal of Mathematics
Hence, by (33), (34), and (39), we obtain the following theorem which shows the relationship between degenerate Daehee polynomials of the second kind and degenerate Bernoulli polynomials of the second kind.
Theorem 2. For nonnegative integer n and d ∈ N with d ≡ 1(mod 2), we have By (25), we note that By comparing the coefficients on both sides of (41), we obtain the following theorem.

Theorem 3. For nonnegative integer n, we have
Note that if we put f(x) � (1 + λ log(1 + t)) x/λ , then and thus, by (3), we have Moreover, and, by (43), we obtain Journal of Mathematics From (43), (44), and (46), we obtain the following theorem which represents a recurrence relations between degenerate Daehee polynomials of the second kind and degenerate Daehee numbers of the second kind.

Higher-Order Degenerate Daehee Polynomials of the Second Kind
In this section, we consider the higher-order degenerate Daehee polynomials of the second kind given by the generating function as follows: for the given positive real number r, In particular, if x � 0, D (r) n (0 | λ) � D (r) n (λ) are called the higher-order degenerate Daehee numbers of the second kind.
Note that From (35), we note that In addition, by replacing t by e t − 1 in (49), we have Hence, by (51)-(53), we obtain the following theorem.

Theorem 5.
For n ≥ 0, we have 6 Journal of Mathematics Moreover, In particular, if r � 1, then we know that eorem 5 is a generalization of eorem 1 and eorem 2.
Note that for each k, r ∈ N, It is well known that for each k ∈ Z, By (23), (7), and (57), we have By (56) and (58), we obtain the following theorem.
Theorem 6. For each n, q ≥ 0, we have n (x + 1)S 2,λ (q − l, n)S 1 (q, p). (59) Note that by (3), By (17) and (60), we have us, by (60) and (61), we obtain the following theorem which shows that higher-order degenerate Daehee polynomials of the second kind are represented by linear combination of the higher-order Carlitz's type degenerate Bernoulli polynomials.

Conclusion
In the past two decades, the degenerations of special functions and their applications have been studied as a new area of mathematics. In this paper, we considered the degenerate Daehee numbers and polynomials by using p-adic invariant integral on Z p which are different from Kim's degenerate Daehee polynomials. We derive some new and interesting properties of those polynomials.
Next, from the definition of the higher-order degenerate Daehee numbers and Daehee polynomials of the second kind, we found the relationship between the degenerate Bernoulli polynomials, the first and second Stirling numbers, the Bernoulli polynomials, degenerate Stirling numbers of the second kind, and those numbers and polynomials.

Data Availability
No data were used to support this study.