Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the ﬁ rst kind. Moreover, we de ﬁ ne degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

It is readily seen that Recently, Kim et al. [9] introduced the modified degenerate polyexponential function which is defined by the following: Here and in the following, let ℤ denote the set of integers. We note that The degenerate poly-Genocchi polynomials are defined as follows (see [9]): Letting ζ = 0, G ðkÞ j,λ = G ðkÞ j,λ ð0Þ are called the poly-Genocchi numbers.
For i ≥ 0, the degenerate Stirling numbers of the first kind are defined by means of the following generating function (see [4]): Note that lim λ⟶0 S 1,λ ðj, kÞ = S 1 ðj, kÞ are the Stirling numbers of the first kind given by the following (see [3,18]): For i ≥ 0, the degenerate Stirling numbers of the second kind are defined by means of the following generating function (see [18]): We note that lim λ⟶0 S 2,λ ðj, kÞ = S 1 ðj, kÞ are the Stirling numbers of the second kind given by the following (see [3][4][5][6][7]18]): The subsequent content of this paper is organized as follows: In Section 2, we define the degenerate poly-Frobenius-Euler polynomials and numbers by using the modified degenerate polyexponential functions and derive some properties and relations of these polynomials. In Section 3, we consider the degenerate poly-Frobenius-Euler polynomials of a complex variable and then we derive several properties and relations. Also, we examine the results derived in this study.

Degenerate Poly-Frobenius-Euler Numbers and Polynomials
In this section, we define degenerate poly-Frobenius-Euler numbers and polynomials and investigate some properties of these polynomials. We begin following the definition as follows.
2 Journal of Function Spaces Definition 1. We consider the degenerate poly-Frobenius-Euler polynomials are defined by means of the following generating function: where λ, u ∈ ℂ with u ≠ 1 and k ∈ ℤ.
Theorem 2. Let j ≥ 0. Then, we have the following: Proof. Using Equation (16), we see that On the other hand, In view of Equation (22), we complete the proof. ☐ Theorem 3. Let j ≥ 0. Then, we have the following: Proof. In Equation (16), we observe that By Equations (16) and (24), we require at the desired result. ☐

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Proof. By using Equations (14) and (16), we see that In view of Equation (26), we complete the proof. ☐

Corollary 5.
For k ∈ ℤ and j ≥ 0. Then, Corollary 6. For j ≥ 0. Then, Corollary 7. On setting u = −1 and k = 1 and using Equation (4), Theorem 3 to get It is well known from [7] that where B ðrÞ j ðζÞ are called the higher-order Bernoulli polynomials which are given by the generating function (see [3,16]): Theorem 8. For j ≥ 0. Then, we have the following: Proof. In Equation (8), we note that From Equation (33), for k ≥ 1, we have the following: For k ≥ 2 in the above expression, we have the following: In view of Equations (35) and (36), we obtain at the desired result. ☐ Theorem 9. Let j ≥ 0. Then, we have the following:

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Proof. Using Equation (16), we get the following: Thus, by Equation (38), we complete the proof. ☐ Theorem 10. Let j ≥ 0. Then, we have the following: Proof. In Equation (16), we see that Comparing the coefficients of z j on both sides, we get the result. ☐ Theorem 11. Let j ≥ 0. Then, Proof. From Equation (16), we see that By Equation (42). We complete the proof. ☐ Theorem 12. Let j ≥ 0. Then, Proof. By changing ζ by ζu + α in Equation (16), we get the following: Therefore, by Equations (16) and (44), we obtain the result. ☐

Degenerate Unipoly-Frobenius-Euler Numbers and Polynomials
In this section, we introduce degenerate unipoly-Frobenius-Euler polynomials by using degenerate unipoly polynomials and derive some important properties of these polynomials. In [3], Kim and Kim introduced unipoly function. In the view of [9], the degenerate unipoly function is defined by Dolgy and Khan [19] as follows: Note that, we have the following: is the modified degenerate polylogarithm function. It is clear that are called the unipoly function attached to polynomials pðζÞ (see [20]).

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From Equation (47), we have the following: is the ordinary polylogarithm function. By using Equations (45) and (16), we define the degenerate unipoly-Frobenius-Euler polynomials as follows: At the special value ζ = 0, ℍ Proof. Let us take pðjÞ = 1/Γλ. Then, we have the following: Therefore, by Equations (49) and (53), we get the result. ☐ Corollary 15. Let j ≥ 0. Then, we have the following: