On a new type of degenerate poly-Fubini numbers and polynomials

. In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.


Introduction
Carlitz's [2,3] initiated a study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers. Kim and Kim et al. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] have studied the degenerate versions of special numbers and polynomials actively. This idea provides a powerful tool to define special numbers and polynomials of their degenerate versions. We can say that the notion of degenerate version forms a special class of polynomials because of their great applicability. The most important applications of these polynomials are in the theory of finite differences, analytic number theory, application in classical analysis, and statistics. Despite the applicability of special functions in classical analysis and statistics, they also arise in communications systems, quantum mechanics, nonlinear wave propagation, electric circuit theory, electromagnetic theory, etc.
It is well known, that the two-variable Fubini polynomials are defined by means of the following generating function (see [29]) (1.1) When x = 0 in (1.1), the two variable Fubini polynomials F n (x, y) reduce to the usual Fubini polynomials given by F n (y) t n n! , (see [1,4,7]). (1.2) It is easy to see that For y = 1 in (1.2), we get the familiar Fubini numbers F n (1) = F n as follows F n t n n! , (see [37]). (1.4) For more information about the applications of the usual Fubini polynomials and numbers (see [5,11,12,34,35]).
In the case when x = 0, B j,λ = B j,λ (0) are called the degenerate Bernoulli numbers and x = 0, E j,λ = E j,λ (0) are called the degenerate Euler numbers.
For s ∈ Z, the polylogaritm function is defined by a power series in z as [8,32]). (1.8) It is notice that For λ ∈ R, Kim-Kim [26] defined the degenerate version of the logarithm function, denoted by log λ (1 + z) as follows: [11]) (1.10) being the inverse of the degenerate version of the exponential function e λ (z) as has been shown below e λ (log λ (z)) = log λ (e λ (z)) = z.
It is noteworthy to mention that The degenerate polylogarithm function [26] is defined by Kim-Kim to be It is clear that [3,4]).
From (1.10) and (1.11), we get Very recently, Kim-Kim [26] introduced the new type degenerate version of the Bernoulli polynomials and numbers, by using the degenerate polylogarithm function as follows j,λ (0) are called the degenerate poly-Bernoulli numbers.
Lee-Kim-Jang [33] introduced the type 2 degenerate poly-Euler polynomials and numbers as follows n,λ (0) are called the type 2 degenerate poly-Euler numbers.
The degenerate Stirling numbers of the second kind [16] are given by standing for the Stirling numbers of the second kind given by means of the following generating function: ).
In this paper, we construct the new type of degenerate poly-Fubini polynomials and numbers by using degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini numbers and polynomials. Furthermore, we introduce the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using degenerate polylogarithm function and investigate some properties for those polynomials. Also, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

A new type of degenerate poly-Fubini numbers and polynomials
In this section, we define the new type of degenerate poly-Fubini numbers and polynomials by using the degenerate polylogarithm function which are called the new type of degenerate poly-Fubini numbers and polynomials and investigate some properties of these polynomials. We start following definition as follows. Now, we consider the new type of 2 variable degenerate poly-Fubini polynomials which are called the new type of 2 variable degenerate poly-Fubini polynomials and given by Obviously, we have Therefore, by equation (2.3), we obtain the following theorem.
In general, from (2.8), we note that Therefore by comparing the coefficients on both sides of (2.7), we obtain the following theorem.
Theorem 2.5. Let k ∈ Z and n ≥ 0, we have .
For k = 1 in Theorem 2.5., we get the following corollary  .

From (2.1), we note that
t n n! .
Theorem 2.6. Let k ∈ Z and n ≥ 0, we have When x = 0 and x = −1 in Theorem (2.6), we get On the other hand, ) t n n! . (2.21) Therefore, by (2.20) and (2.21), we get the following theorem.
Comparing the coefficients on both sides of (2.22), we obtain the following theorem.
Theorem 2.8. Let k ∈ Z and n ≥ 0, we have On setting x = 0 in Theorem 2.8, we get Corollary 2.5. Let k ∈ Z and n ≥ 0, we have ) t n n! .

New type degenerate unipoly-Fubini numbers and polynomials
In this section, we introduce new type of degenerate unipoly-Fubini polynomials by arising from degenerate polylogarithm function and derive several properties of these polynomials.
Let p be any arithmetic function which is a real or complex valued function defined on the set of positive integers N. Kim-Kim [15] defined the unipoly function attached to polynomials p(x) by is the ordinary polylogaritm function.
In this paper, we define the degenerate unipoly function attached to polynomials p(x) as follows: It is worthy to note that is the degenerate polylogarithm function. Now, we define the new type of degenerate unipoly-Fubini polynomials attached to polynomials p(x) by In the case when n,λ,p (0) are called the new type degenerate unipoly-Fubini numbers attached to p.

Conclusion
In this paper, we introduced the new type of degenerate poly-Fubini polynomials and numbers by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini numbers and polynomials. We represented the generating function of the degenerate poly-Fubini numbers by iterated integrals in Theorem 2.2, 2.3, and 2.4 and explicit degenerate poly-Euler polynomials in terms of the Bernoulli polynomials and degenerate Stirling numbers of the second kind in Theorem 2.5. We also represented those numbers in terms of the degenerate Stirling numbers of the second kind in Theorem 2.6. In the final section, we defined the degenerate unipoly-Fubini polynomials by using the degenerate polylogarithm function and obtained the identity degenerate unipoly-Fubini polynomials in terms of the degenerate Fubini numbers and degenerate Stirling numbers of the second kind in Theorem 3.1 and Theorem 3.2, the degenerate unipoly-Fubini numbers and the Stirling numbers of the first kind in Theorem 3.3.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.