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

Abstract. In [18], Kim et al. introduced the degenerate poly-Bernoulli polynomials by using polyexponential function. In this paper, we study the degenerate poly-Frobenius-Genocchi polynomials, which are called the type 2 degenerate polyFrobenius-Genocchi polynomials, by means of polyexponential function. Then, we derive some useful relations and properties. We derive type 2 degenerate poly-FrobeniusGenocchi polynomials equal a linear combination of the degenerate Frobenius-Genocchi polynomials and Stirling numbers of the first kind. Furthermore, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.


Introduction
Recently, Kim and his research team (see [7][8][9][10][11][12][13][14][15][16][17][18]) 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 from a special class of polynomials because of their great applicability. The most important application of these polynomials is in the 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.
As is well known, the classical Bernoulli, Euler and Genocchi polynomials are respectively, defined by (see [4,5,6]) and In the case when x = 0, B j = B j (0), E j = E j (0) and G j = G j (0) are respectively, called the Bernoulli, Euler and Genocchi numbers.
For u ∈ C with u ̸ = 1, the Frobenius-Genocchi polynomials G F n (x; u) are defined by t n n! , (see [18). (1.2) In the case when x = 0, G F n (u) = G F n (0; u) are called the Frobenius-Genocchi numbers.
Obviously G F n (x; −1) = G n (x). For k ∈ Z, Kim-Kim [7] defined the modified polyexponential function, as an inverse to the polylogarithm function by It is worthy to note that e(x, 1|k) = 1 x Ei k (x) and Ei 1 (x) = e x − 1.

Note that lim
In [17], Kim et al. considered the the degenerate Genocchi polynomials given by 2z In the case when u = 0, G j,λ = G j,λ (0) are called the degenerate Genocchi numbers.
Very recently, Kim et al. [18] introduced the degenerate poly-Bernoulli polynomials defined by n,λ (0) are called the degenerate poly-Bernoulli numbers.

Type 2 degenerate poly-Frobenius-Genocchi polynomials
Let λ, u ∈ C with u ̸ = 1 and k ∈ Z, by using the polyexponential function, we consider the type 2 degenerate poly-Frobenius-Genocchi polynomials are defined by means of the following generating function In case when x = 0 in (2.1), G n,λ (0; u) are called the type 2 degenerate poly-Frobenius-Genocchi numbers.
For k = 1 in (2.1), we get where G F n,λ (x; u) are called the degenerate Frobenius-Genocchi polynomials.
Therefore, by (2.1) and (2.9), we require at the desired result.
(2.11) By comparing the coefficients of t n n! , we arrive at the desired result (2.10). Corollary 2.1. For k ∈ Z and n ≥ 0, we have
By (15) and (16), we require at the desired result. Thus, we complete the proof.
Proof. Let k ≥ 1, be an integer. For s ∈ C, we define the function χ k,ν (s) as 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 (2.18), 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, 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 (2.20) and (2.21), we obtain . Thus, we complete the proof of this theorem.
By comparing the coefficients of T n , we obtain the result (2.23).

Theorem 2.7.
For n ≥ 0, we have Comparing the coefficients of t n on both sides, we get the result (2.25).

Type 2 degenerate unipoly-Frobenius-Genocchi 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 [7] defined the unipoly function attached to polynomials p(x) by Moreover, x n n k = Li k (x), (see [3]), (3.2) is the ordinary polylogaritm function.
(3.7) Therefore, by comparing the coefficients on both sides of (3.7), we obtain the following theorem.
Theorem 3.2. Let n ≥ 0 and k ∈ Z. Then we have By comparing the coefficients of t n , we obtain the result (3.8). By comparing coefficients on both sides of (3.11), we obtain the following theorem.

Conclusions
Motivated by the definition of the type 2 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 Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Genocchi 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-Genocchi polynomials equal a linear combination of the degenerate Frobenius-Genocchi polynomials and degenerate Stirlings numbers of the first and second kind. In a special case, we have given a relation between the type 2 degenerate Frobenius-Genocchi polynomials and Bernoulli polynomials of order n. Moreover, inspired by the definition of unipoly-Bernoulli polynomials introduced by Kim-Kim [7] we have introduced the type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and given multifarious properties including degenerate Stirling numbers of the second kind and degenerate Frobenius-Genocchi polynomials.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.