A note on type 2 degenerate poly-Frobenius-Euler polynomials

Abstract. In this paper, we construct the degenerate poly-Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of polyexponential function. We derive explicit expressions and some identities of those polynomials. In the last section, we introduce type 2 degenerate unipoly-FrobeniusGenocchi polynomials by means of unipoly function and derive explicit multifarious properties.


Introduction
Special polynomials and their generating functions have important roles in many branches of mathematics, probability, statistics, mathematical physics, and also engineering. Since polynomials are suitable for applying well-known operations such as derivative and integral, polynomials are very useful to study real-world problems in the aforementioned areas. For instance, generating functions for special polynomials with their congruence properties, recurrence relations, computational formulae, and symmetric sum involving these polynomials have been many authors in recent years (see ).
Recently, Kim and his research team (see [13][14][15][16][17][18][19]) have studied the degenerate versions of special numbers and polynomials actively. This idea provides a powerfool tool in order to define special numbers and polynomials of their degenerate versions. We can say that the notion of degenerate version from a special class of polynomials because of their great applicability. The most important of application of these polynomials are in theory of finite differences, analytic number theory, applications 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.
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 [6,7,19,20,23]) (1.1) respectively.
In (2017), Kurt [10] introduced the poly-Frobenius-Euler polynomials are given by In the case when n (0; u) are called the poly-Frobenius-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 lim λ−→0 h n,λ (x|u) = H n (x|u), (n ≥ 0). [13] introduced the polyexponential function, as an inverse to the polylogarithm function to be
It is well known that the Stirling numbers of the first kind are defined by (x) n = n ∑ l=0 S 1 (n, l)x l , (see [24,25]), (1.11) where (x) 0 = 1, and (x) n = x(x − 1) · · · (x − n + 1), (n ≥ 1). From (1.11), it is easily to see that [13,18]). (1.12) In the inverse expression to (1.12), the Stirling numbers of the second kind are defined by [10,12,22]). (1.13) From (1.13), it is easily to see that [22,24,25]). (1.14) 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 type 2 oly-Frobenius-Euler polynomials and numbers by using the polyexponential function and derive several properties on the degenerate poly-Frobenius-Euler polynomials and numbers. In the final section, we define type 2 unipoly-Frobenius-Euler polynomials by means of unipoly function and derive explicit expressions of those polynomials.

Type 2 degenerate poly-Frobenius-Euler polynomials
Let λ, u ∈ C with u ̸ = 1 and k ∈ Z, by using the polyexponential function, we consider the type 2 degenerate poly-Frobenius-Euler polynomials are defined by means of the following generating function In the special case, n,λ (0; u) are called the type 2 degenerate poly-Frobenius-Euler numbers.
Therefore, by (2.1) and (2.6), we require at the desired result.
(2.7) By comparing the coefficients of t n n! , we complete the proof. Corollary 2.1. For k ∈ Z and n ≥ 0, we have
(2.11) By (2.10) and (2.11), we require at the desired result. Thus, we complete the proof.
Ei k (log(1 + (1 − u)z)) dz. (2.12) In view of calculation above that χ k,ν (s) is holomorphic function for ℜ(s) > 0 because of the comparison test as Ei k (log(1 + (1 − u)z)) ≤ E1 k (log(1 + (1 − u)z)) with the assumption (1 − u)t ≥ 0. From (2.11), we note that 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 ℜ(s) > 0, the first integral in (2.13) can be written as which defines an entire function of s. Thus, we may include that χ k,ν (s) can be continued to an entire function of s. Further, from (2.14) and (2.15), we obtain Proof. From (2.1), we have Comparing the coefficients on both sides, we get the result. Proof. By (2.1), we observe that Comparing the coefficients of t n on both sides, we get the .
By comparing the coefficients of t n on both sides, we get the result. Proof. Replacing x by x + α in (2.1), we have x m m!S 2,λ (l + α, m + α)H

Type 2 degenerate unipoly-Frobenius-Euler 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 [13] defined the unipoly function attached to polynomials p(x) by x n n k = Li k (x), (see [4,9]), (3.2) is the ordinary polylogaritm function.
(3.8) Therefore, by comparing the coefficients on both sides of (3.8), we obtain the result.
Proof. Using (3.3), we observe that ) t n n! . (3.13) By comparing the coefficients of t n on both sides, we get the result. Proof. In order to prove that, we observe that By comparing coefficients on both sides of (3.15), we obtain the result.

Conclusions
Motivated by the definition of the degenerate poly-Bernoulli polynomials introduced by Kim et al. [18], in the present paper, we have considered a class of new generating function for the degenerate Frobenius-Euler polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of the polyexponential function. Then, we have derived some useful relations and properties. We have showed that the type 2 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 type 2 degenerate Frobenius-Euler polynomials and Bernoulli polynomials of order n. Moreover, inspired by the definition of unipoly-Bernoulli polynomials introduced by Kim-Kim [] we have introduced the type 2 degenerate unipoly-Frobenius-Euler polynomials by means of unipoly function and given multifarious properties including degenerate Stirling numbers of the second kind and degenerate Frobenius-Euler polynomials.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.