A Note on Type-Two Degenerate Poly-Changhee Polynomials of the Second Kind

: In this paper, we ﬁrst deﬁne type-two degenerate poly-Changhee polynomials of the second kind by using modiﬁed degenerate polyexponential functions. We derive new identities and relations between type-two degenerate poly-Changhee polynomials of the second kind. Finally, we derive type-two degenerate unipoly-Changhee polynomials of the second kind and discuss some of their identities.


Introduction
As is well known, Changhee polynomials Ch n (x) are defined by means of the following generating function Ch n (x) t n n!
(see [1,2]). In the case when x = 0, Ch n (0) = Ch n are called Changhee numbers. The Euler polynomials are defined by the following generating function: Z p e (x+y)t d µ −1 y = 2 e t + 1 e xt = ∞ ∑ n=0 E n (x) t n n!
In [9], Kim et al. introduced degenerate poly-Genocchi polynomials, which are given by In the case when n,λ are called degenerate poly-Genocchi numbers. Let λ ∈ C p with | λ |≤ 1. The degenerate Changhee polynomials of the second kind Ch n,λ (x) are defined by (see [10]). When x = 0, Ch n,λ (0) = Ch n,λ are called the degenerate Changhee numbers of the second kind.
In this article, we introduce type-two degenerate poly-Changhee polynomials of the second kind and derive explicit expressions and some identities of those polynomials. In addition, we introduce type-two degenerate unipoly-Changhee polynomials of the second kind and derive explicit multifarious properties.

Type-Two Degenerate Poly-Changhee Polynomials of the Second Kind
In this section, we define degenerate Changhee polynomials of the second kind by using the modified degenerate polyexponential function; these are called type-two degenerate poly-Changhee numbers and polynomials of the second kind in the following.
Let λ ∈ C and k ∈ Z; we consider that the type-two degenerate poly-Changhee polynomials of the second kind are defined by In the special case, when x = 0, Ch n,λ are called type-two degenerate poly-Changhee numbers of the second kind, where log λ (t) = 1 λ (t λ − 1) is the compositional inverse of e λ (t) that satisfies log λ (e λ (t)) = e λ (log λ (t)) = t.
For k = 1 in (16), we get where Ch n,λ (x) are called degenerate Changhee polynomials of the second kind (see Equation (10)). Obviously, where Ch n (x) are called type-two poly-Changhee polynomials.
Corollary 1. For n ≥ 0 and k ∈ Z, we have From (16), we observe that By comparing the coefficients on both sides of (22), we obtain the following theorem.
Theorem 2. Let n ≥ 0 and k ∈ Z. Then, we have In [4], the degenerate Bernoulli polynomials of the second kind are defined by (see [30]). For x = 0, b n,λ (0) = b n,λ are called degenerate Bernoulli numbers of the second kind. From (7), we note that Thus, from (16) and (25), we have Therefore, using (26), we obtain the following theorem. Let k ≥ 1 be an integer. For s ∈ C, we define the function η k,λ (s) as The second integral converges absolutely for any s ∈ C, and hence, the second term on the right-hand side vanishes at non-positive integers. That is, On the other hand, for (s) > 0, the first integral in (29) can be written as which defines an entire function of s. Thus, we may conclude that η k,λ (s) can be continued to an entire function of s. Further, from (28) and (29), we obtain Therefore, using (30), we obtain the following theorem. From (16), we note that On the other hand, Therefore, using (31) and (32), we obtain the following theorem. (1) m,λ S 1,λ (n, m) For k = 1 in Theorem 5, we get the following corollary. From (16), we note that By comparing the coefficients on both sides of (33), we get the following theorem.

Type-Two Degenerate Unipoly-Changhee Polynomials of the Second Kind
The unipoly function u k (x|p) is defined by Kim and Kim to be (see [20]): where p is any arithmetic function that is a real or complex valued function defined on the set of positive integers N. Moreover, (see [22,23,28]) is the ordinary polylogarithm function.
In this paper, we consider the degenerate unipoly function attached to polynomials p(x) as follows: It is worth noting that is the modified degenerate polyexponential function. By using (36), we define type-two degenerate unipoly-Changhee polynomials of the second kind by 2u k,λ (log λ (1 + t)|p) In the case when x = 0, Ch n,λ,p are called type-two degenerate unipoly-Changhee numbers of the second kind. Let us take p(n) = 1 Γλ . Then, we have Thus, using (39), we have the following theorem.
Theorem 8. Let n ∈ N and k ∈ Z. Then, we have In particular, From (38), we observe that From (44), we obtain the following theorem.
Theorem 9. Let n ≥ 0 and k ∈ Z. Then, we have From (38), we observe that By comparing the coefficients on both sides of (46), we obtain the following theorem.

Conclusions
In this article, we introduced type-two degenerate poly-Changhee polynomials of the second kind and derived some beautiful identities and relations between type-two degenerate poly-Changhee numbers of the second kind and Stirling numbers of first and second kind. In addition, we gave the relation between degenerate Bernoulli polynomials of the second kind and type-two degenerate poly-Changhee numbers of the second kind. Again, we defined type-two degenerate unipoly-Changhee polynomials of the the second kind and obtained some properties and relationships of degenerate unipoly-Changhee numbers of the second kind and the Daehee numbers.
Author Contributions: Both authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Both authors have read and agreed to the published version of the manuscript.