On the new type of degenerate poly-Genocchi numbers and polynomials

Kim and Kim (J. Math. Anal. Appl. 487:124017, 2020) introduced the degenerate logarithm function, which is the inverse of the degenerate exponential function, and defined the degenerate polylogarithm function. They also studied a new type of the degenerate Bernoulli polynomials and numbers by using the degenerate polylogarithm function. Motivated by their research, we subdivide this paper into two parts. In Sect. 2, we construct a new type of degenerate Genocchi polynomials and numbers by using the degenerate polylogarithm function, called the degenerate poly-Genocchi polynomials and numbers, deriving several combinatorial identities related to the degenerate poly-Genocchi numbers and polynomials. Then, in Sect. 3, we also consider the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. In particular, we provide some new explicit computational identities of degenerate unipoly polynomials related to special numbers and polynomials.

In 1997, Kaneko [2] introduced poly-Bernoulli numbers which are defined by the polylogarithm function. The polyexponential functions were first studied by Hardy [30] and reconsidered by Kim and Kim [1,17] in view of an inverse to the polylogarithm functions which were studied by Jaonquière [31], Lewis [8], and Zagier [32]. Kim et al. [18] also studied a new type of the degenerate poly-Bernoulli polynomials by using the degenerate modified polyexponential functions.
Furthermore, Kim and Kim [15] introduced the degenerate logarithm function (the inverse of the degenerate exponential function) and studied a new type of the degenerate Bernoulli polynomials and numbers by using the degenerate polylogarithm function. Influenced by Kim et al. 's research, as well as the importance and potential for applications in number theory, combinatorics, and other fields of applied mathematics, we define a new type of the degenerate poly-Genocchi polynomials and the degenerate unipoly Genocchi polynomials, and provide several combinatorial identities related to these polynomials and numbers. Now, as is well established in academia, the ordinary Bernoulli polynomials B n (x) and the Genocchi polynomials G n (x), (n ∈ N ∪ {0}) are respectively defined by their generating functions as follows (see [9,13,14,20]): When x = 0, B n = B n (0) and G n = G n (0) are respectively called the Bernoulli numbers and the Genocchi numbers.
This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1: it can be extended to |z| ≥ 1 by analytic continuation.
From (9), we note that Recently, Kim and Kim [15] introduced the degenerate logarithm function log λ (1 + t), which is the inverse of the degenerate exponential function e λ (t) and the motivation for the definition of degenerate polylogarithm function as follows: Here, log λ (t) = 1 λ (t λ -1) is the compositional inverse of e λ (t) satisfying log λ (e λ (t) = e λ (log λ (t)) = t. We note that Thus, the degenerate polylogarithm function is defined by (-λ) n-1 (1) n,1/λ (n -1)!n k x n , k ∈ Z |x| < 1 , see [15] . (13) We note that By (11) and (13), we see that In [15], they also studied a new type of degenerate poly-Bernoulli polynomials and numbers by using the degenerate polylogarithm function as follows: Kim [15] introduced the degenerate Stirling numbers of the second kind as follows: As an inversion formula of (18), the degenerate Stirling numbers of the first kind are defined by From (18) and (19), it is well known that and [15] .
This paper is subdivided into two parts. In Sect. 2, we construct a new type of degenerate Genocchi polynomials and numbers, called the degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, deriving several combinatorial identities related to the degenerate poly-Genocchi numbers and polynomials.
In Sect. 3, we also consider the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. In particular, we provide some new explicit computational identities of degenerate unipoly polynomials related to special numbers and polynomials.

A new type degenerate poly-Genocchi numbers and polynomials
In this section, we define the new type degenerate poly-Genocchi polynomials by using the degenerate polylogarithm function which are called the degenerate poly-Genocchi polynomials as follows: When n,λ (0) are called the degenerate poly-Genocchi numbers. When k = 1, from (15), we see that g (1) n,λ (x) = G n,λ (x) (n ≥ 0) are the degenerate Genocchi polynomials because of From (5), we observe that Theorem 1 For n ≥ 2, k ∈ Z, we have Proof By using (25), we have and Thus, from (15), (26), and (27), we get the desired result as follows: Theorem 2 For n ≥ 0 and k = 2, we have Proof By using (16), (17), and Theorem 1, we get ∞ n=0 g (2) n,λ Therefore, by comparing the coefficients on both sides of (30), we get what we wanted.
Therefore, by comparing the coefficients on both sides of (32), we get what we desired.
The following lemma is easily obtained by (5) and (22).
Therefore, by comparing the coefficients on both sides of (38), we have the desired result.
Therefore, by comparing the coefficients on both sides of (40), we have the desired result.
Proof From (8), (13), and (36), we have Therefore, by comparing the coefficients on both sides of (42), we have the desired result.

Theorem 9
For n ≥ 1, k ∈ Z, we have Proof From (5), (7), and (36), we get Therefore, by comparing the coefficients on both sides of (44), we have the desired result.
Theorem 10 For n ≥ 1 and k ∈ Z, we get Proof Replace t by -1 2 log λ (1t) in (22). From (7), (13), and (21), the left-hand side of (22) is On the other hand, the right-hand side of (22) is Therefore, by comparing the coefficients of (46) and (47), we get what we wanted.

The degenerate unipoly Genocchi polynomials and numbers
Let p be any arithmetic function which is real-or complex-valued function defined on the set of positive integers N. Kim and Kim [5] defined the unipoly function attached to p(x) Moreover, is the ordinary polylogarithm function.
In this paper, we define the degenerate unipoly function attached to p(x) as follows: We note that is the degenerate polylogarithm function. We also define the degenerate unipoly Genocchi polynomials by When x = 0, g (k) n,λ,p = g (k) n,λ,p (0) is the degenerate unipoly Genocchi numbers. We note that The next lemma is intended to be used conveniently to prove some of the theorems below.
Proof From (7) and Lemma 11, we have Thus, by comparing the coefficients on both sides of (57), we have the desired result.

Conclusion
In this paper, we constructed a new type degenerate Genocchi polynomials and numbers by using the degenerate polylogarithm function, called degenerate poly-Genocchi polynomials and numbers. We represented the following: the generating function of the degenerate poly-Genocchi numbers by iterated integrals in Theorem 1; the explicit degenerate poly-Genocchi numbers in terms of the degenerate Bernoulli polynomials and the degenerate Genocchi numbers in Theorem 3. Not to mention, we obtained in Theorems 5 and 7 that the degenerated poly-Genocchi polynomials are represented by the degenerated poly-Genocchi numbers and the degenerate Stirling numbers of the second kind. We also demonstrated in Theorem 10 that the degenerate poly-Genocchi polynomials are represented by the degenerated poly-Genocchi numbers and the degenerate Stirling numbers of the first kind. We expressed those polynomials in terms of the degenerate Genocchi polynomials and the degenerate Stirling numbers of the second kind in Theorem 8 and the degenerate poly-Bernoulli polynomials in Theorem 9.
On the other hand, in Sect. 3, we defined the degenerate unipoly Genocchi polynomials by using the degenerate polylogarithm function and obtained: the identity degenerate unipoly Genocchi polynomials in terms of the degenerate Euler numbers and the degenerate Stirling numbers of the second kind in Theorem 12; the degenerate Genocchi numbers and the degenerate Stirling numbers of the second kind in Theorem 13; the degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind in Theorem 14.
The field of degenerate versions is widely applied not only to number theory and combinatorics but also to symmetric identities, differential equations, and probability theory. As one of our future projects, we would like to continue to study degenerate versions of certain special polynomials and numbers and their applications to physics, economics, and engineering as well as mathematics.