New Type of Degenerate Poly-Frobenius-Euler Polynomials and Numbers

Abstract. Motivated by Kim-Kim [19] introduced the new type of degenerate polyBernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

The classical Bernoulli B n (x), Euler E n (x) and Genocchi G n (x) polynomial are defined by means of the following generating function as follows and 2t e t + 1 e xt = ∞ ∑ n=0 G n (x) t n n! , | t |< π, (see ) (1.1) respectively.
In (2017), Kurt [10] introduced the poly-Frobenius-Euler polynomials are given by In the case when x = 0, H (k) n (0; u) are called the poly-Frobenius-Euler numbers.
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.
Kim et al. [15] introduced the degenerate Frobenius-Euler polynomials are defined by means of the generating function as follows At the value x = 0, h n,λ (u) = h n,λ (0|u) are called the degenerate Frobenius-Euler numbers.
It is readily seen that For s ∈ Z, the polylogaritm function is defined by a power series in z as [4,9]). (1.8) It is notice that For λ ∈ R, Kim-Kim [19] defined the degenerate version of the logarithm function, denoted by log λ (1 + z) as follows: [18]) (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 [19] is defined by Kim-Kim to be It is clear that , (see [4,9]).
From (1.10) and (1.11), we get Very recently, Kim-Kim [19] introduced the new type degenerate version of the Bernoulli polynomials and numbers, by using the degenerate polylogarithm function as follows In the special case j,λ (0) are called the degenerate poly-Bernoulli numbers.
It is well known that the Stirling numbers of the first kind are defined by [24,25]), (1.13) where ( [12,13,18]). (1.14) In the inverse expression to (1.14), the Stirling numbers of the second kind are defined by [10,12,22] [22,24,25] For n ≥ 0, the degenerate Stirling numbers of the second kind [7,8,17] are defined by In this paper, we construct the degenerate poly-Frobenius-Euler polynomials and numbers, called the new type of poly-Frobenius-Euler polynomials and numbers by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Frobenius-Euler polynomials and numbers.

New type of degenerate poly-Frobenius-Euler polynomials
Let λ, u ∈ C with u ̸ = 1 and k ∈ Z, by using the degenerate polylogarithm function, we define the new type of degenerate poly-Frobenius-Euler polynomials as follows In the special case, n,λ (0; u) are called the new type of degenerate poly-Frobenius-Euler numbers.
where h n,λ (x; u) are called the degenerate Frobenius-Euler polynomials.
Therefore, by (2.1) and (2.3), we require at the desired result.

Theorem 2.4.
For n ≥ 0, we have Proof. Using (1.11), we first consider the following expression By (2.9), we obtain at the desired result. Thus, we complete the proof.
Theorem 2.6. Let k ∈ Z and n ≥ 0, we have Proof. In general, from (2.9), we note that x n n! . (2.11) Therefore, by comparing the coefficients of t n on both sides, we obtain the result. Proof. From (2.1), we have Comparing the coefficients on both sides of (2.12), we get the result.
Proof. By (2.1), we observe that Comparing the coefficients of t n on both sides, we obtain the result. Proof. From (2.1), we have ) t n n! . (2.14) By comparing the coefficients of t n on both sides, we get the result.   ) t n n! .

Conclusions
Motivated by the definition of the degenerate poly-Bernoulli polynomials introduced by Kim-Kim [19], in the present paper, we have considered a class of new generating function for the degenerate Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we have derived some useful relations and properties. We have showed that the new type of degenerate poly-Frobenius-Euler polynomials equal a linear combination of the degenerate Frobenius-Euler polynomials and Stirlings numbers of the first and second kind. In a special case, we have given a relation between the new type of degenerate Frobenius-Euler polynomials and Bernoulli polynomials of order n.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.