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

Authors: Ghulam Muhiuddin; Waseem Ahmad Khan; Deena Al-Kadi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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 first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.


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## Body

## 1. Introduction

Recently, many mathematicians, namely, Carlitz [1, 2], Kim and Kim [3–5], Kim et al. [6–9], Muhiuddin et al. [10–12], and Sharma et al. [13–15] have introduced and studied various degenerate versions of special polynomials and numbers like degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Fubini polynomials, and degenerate Stirling numbers of the first and second kinds.The classical Frobenius-Euler polynomialsℍnαx;u (u∈ℂ with u≠1) of order α are defined by means of the following generating function (see [16, 17]):
(1)1−uez−uαeζz=∑j=0∞ℍjαζ;uzjj!.At the pointζ=0, ℍjαu=ℍjα0;u are called jth Frobenius-Euler numbers of order α.The poly-Frobenius-Euler polynomials due to Kurt [16] are defined as follows:
(2)1−uLik1−e−zzez−ueζz=∑j=0∞ℍjkζ;uzjj!.Whenζ=0, ℍjku=ℍjk0;u are called the poly-Frobenius-Euler numbers.For anyλ∈ℝ (or ℂ), ℝ and ℂ being, respectively, the sets of real numbers and complex numbers, degenerate version of the exponential function eλζz is defined as follows (see [3, 4, 6, 18]):
(3)eλζz≔1+λzζ/λ=∑j=0∞ζj,λzjj!,where ζ0,λ=1 and ζj,λ=ζζ−λ⋯ζ−j−1λ for j≥1 (see [1, 2, 4–10, 18]). It follows from Equation (3) that limλ⟶0eλζz=eζz. Note that eλ1z≔eλz.Carlitz [1, 2] introduced the degenerate Euler polynomials as follows:
(4)2eλz+1eλxz=∑j=0∞Ej,λxzjj!.In the case whenζ=0, Ej,λ=Ej,λ0 are called the degenerate Euler numbers.Note that(5)limλ⟶0Ejζ;λ=Ejζ.The degenerate Frobenius-Euler polynomials are defined by the following (see [6]):
(6)1−u1+λz1/λ−u1+λzζ/λ=∑j=0∞hj,λζ∣uzjj!.At the valueζ=0, hj,λu=hj,λ0∣u are called the degenerate Frobenius-Euler numbers.It is readily seen that(7)limλ⟶0hj,λζ∣u=ℍjζ∣u,j≥0.Recently, Kim et al. [9] introduced the modified degenerate polyexponential function which is defined by the following:
(8)Eik,λζ=∑j=1∞1j,λζjj−1!jk,ζ<1,k∈ℤ.Here and in the following, letℤ denote the set of integers.We note that(9)Ei1,λζ=∑j=1∞1j,λζjj!=eλζ−1.The degenerate poly-Genocchi polynomials are defined as follows (see [9]):
(10)2Eik,λlogλ1+zeλz+1eλζz=∑j=0∞Gj,λkζzjj!,k∈ℤ.Lettingζ=0,Gj,λk=Gj,λk0 are called the poly-Genocchi numbers.The degenerate Daehee polynomialsDj,λζ are defined as follows (see [8]):
(11)logλ1+zz1+zζ=∑j=0∞Dj,λζzjj!.Dj,λ=Dj,λ0 are called the degenerate Daehee numbers.Fori≥0, the degenerate Stirling numbers of the first kind are defined by means of the following generating function (see [4]):
(12)1i!logλ1+zi=∑j=i∞S1,λj,izjj!.Note thatlimλ⟶0S1,λj,k=S1j,k are the Stirling numbers of the first kind given by the following (see [3, 18]):
(13)1i!log1+zi=∑j=i∞S1j,izjj!,i≥0.Fori≥0, the degenerate Stirling numbers of the second kind are defined by means of the following generating function (see [18]):
(14)1i!eλz−1i=∑j=i∞S2,λj,izjj!.We note thatlimλ⟶0S2,λj,k=S1j,k are the Stirling numbers of the second kind given by the following (see [3–7, 18]):
(15)1i!ez−1i=∑j=i∞S2j,izjj!,i≥0.The subsequent content of this paper is organized as follows: In Section2, 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.


## 2. 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.Definition 1.
We consider the degenerate poly-Frobenius-Euler polynomials are defined by means of the following generating function:(16)Eik,λlogλ1+1−uzzeλz−ueλζz=∑j=0∞ℍj,λkζ;uzjj!,where λ,u∈ℂ with u≠1 and k∈ℤ.Upon setting,ζ=0, ℍj,λku=ℍj,λk0;u are called the degenerate poly-Frobenius-Euler numbers, where logλz=1/zzλ−1 is the compositional inverse of eλz satisfying
(17)logλeλz=eλlogλz=z.Adjustingk=1 in Equation (16), we get the following:
(18)1−ueλz−ueλζz=∑j=0∞hj,λζ;uzjj!,where hj,λζ;u are called the degenerate Frobenius-Euler polynomials (see [6]).Obviously,(19)limλ⟶0Eik,λlogλ1+1−uzzeλz−ueλζz=∑j=0∞limλ⟶0ℍj,λkζ;uzjj!=Eiklog1+1−uzzez−ueζz=∑j=0∞ℍjkζ;uzjj!,where ℍjkζ;u are called the type 2 poly-Frobenius-Euler polynomials.Theorem 2.
Letj≥0. Then, we have the following:
(20)∑p=0jjp∑s=0p1s+1,λs+1k−1S1,λp+1,s+1ζj−p,λ1−up+1p+1=∑s=0jjsℍj−s,λkζ;u1s,λ−uℍj,λkζ;u.Proof.
Using Equation (16), we see that
(21)Eik,λlogλ1+1−uzzeλζz=eλz∑j=0∞ℍj,λkζ;uzjj!−u∑j=0∞ℍj,λkζ;uzjj!=∑s=0∞1s,λzss!∑j=0∞ℍj,λkζ;uzjj!−u∑j=0∞ℍj,λkζ;uzjj!=∑j=0∞∑s=0jjsℍj−s,λkζ;u1s,λ−uℍj,λkζ;uzjj!.
On the other hand,(22)Eik,λlogλ1+1−uzzeλζz=∑j=0∞ζj,λzjj!1z∑s=1∞1s,λlogλ1+1−uzss−1!sk=∑j=0∞ζj,λzjj!1z∑s=0∞1s+1,λs+1k−1∑l=s+1∞S1,λl,s+11−ulzll!=∑j=0∞ζj,λzjj!∑l=0∞∑s=0l1s+1,λs+1k−1S1,λl+1,s+11−ul+1l+1zll!=∑j=0∞∑p=0jjp∑s=0p1s+1,λs+1k−1S1,λp+1,s+1ζj−p,λ1−up+1p+1zjj!.
In view of Equation (22), we complete the proof.☐Theorem 3.
Letj≥0. Then, we have the following:
(23)ℍj,λkζ;u=∑r=0jjrℍj−r,λkuζr,λ.Proof.
In Equation (16), we observe that
(24)∑j=0∞ℍj,λkζ;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζz=∑j=0∞ℍj,λkuzjj!∑r=0∞ζr,λzrr!=∑j=0∞∑r=0jjrℍj−r,λkuζr,λzjj!.
By Equations (16) and (24), we require at the desired result. ☐Theorem 4.
Letj≥0. Then,
(25)ℍj,λkζ;u=∑q=0jjq∑r=0q1r+1,λr+1kS1q+1,r+11−uqq+1ℍj−q,λζ;u.Proof.
By using Equations (14) and (16), we see that
(26)∑j=0∞ℍj,λkζ;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζz=eλζzzeλz−u∑r=1∞1r,λlogλ1+1−uzrr−1!rk=eλζzzeλz−u∑r=0∞1r+1,λlogλ1+1−uzr+1r!r+1k=eλζzzeλz−u∑r=0∞1r+1,λr+1k∑j=r+1∞S1,λj,r+11−utnn!=1−ueλz−ueλζz∑r=0∞1r+1,λr+1k∑r=j∞S1,λj+1,r+11−ujj+1zjj!=∑j=0∞ℍj,λζ;uzjj!∑l=0∞∑r=0l1r+1,λr+1kS1,λl+1,r+11−ull+1zll!=∑j=0∞∑q=0jjq∑r=0q1r+1,λr+1kS1,λq+1,r+11−uqq+1ℍj−q,λζ;uzjj!.
In view of Equation (26), we complete the proof.☐Corollary 5.
Fork∈ℤ and j≥0. Then,
(27)ℍj,λku=∑q=0jjq∑r=0q1r+1kS1,λq+1,r+11−uqq+1ℍj−q,λu.Corollary 6.
Forj≥0. Then,
(28)ℍj,λkζ;u=∑q=0jjq∑r=0qS1,λq+1,r+11−uqq+1ℍj−q,λζ;u.Corollary 7.
On settingu=−1 and k=1 and using Equation (4), Theorem 3 to get
(29)Ej,λx=∑q=0jjl∑r=0qS1,λq+1,r+12qq+1Ej−q,λζ,j≥0.It is well known from [7] that
(30)zlog1+zr1+zζ−1=∑j=0∞Bjj−r+1ζzjj!,r∈ℂ,where Bjrζ are called the higher-order Bernoulli polynomials which are given by the generating function (see [3, 16]):
(31)zez−1reζz=∑j=0∞Bjrζzjj!.Theorem 8.
Forj≥0. Then, we have the following:
(32)ℍj,λ2u=∑l=0jjl1−ulBlll+1ℍj−l,λu.Proof.
In Equation (8), we note that
(33)ddζEik,λlogλ1+1−ux=ddζ∑j=1∞logλ1+1−uζjj+1!jk=1−u1+1−uζlogλ1+1−uζ∑j=1∞logλ1+1−uζjj+1!jk−1=1−u1+1−uζlogλ1+1−uζEik−1,λlogλ1+1−uζ.
From Equation (33), for k≥1, we have the following:
(34)∑j=0∞ℍj,λkuζjj!=1−uk−1ζeλζ−u∫0ζ11+1−utlogλ1+1−uz×∫0z11+1−uzlogλ1+1−uz⏟k−2−times⋯∫0zz1+1−uzlogλ1+1−uzdzdz⋯dz.
Fork≥2 in the above expression, we have the following:
(35)∑j=0∞ℍj,λ2uζjj!=1−uζeλζ−u∫0ζ1−uz1+1−uzlogλ1+1−uz,(36)∑j=0∞Hj,λuξjj!∑l=0∞1−ulBlll+1ξll!=∑j=0∞∑l=0jjl1−ulBlll+1Hj−l,λuξjj!
In view of Equations (35) and (36), we obtain at the desired result.☐Theorem 9.
Letj≥0. Then, we have the following:
(37)ℍj,λkζ+η;u=∑r=0jjrℍj−r,λkζ;uηr,λ.Proof.
Using Equation (16), we get the following:
(38)∑j=0∞ℍj,λkζ+η;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζ+ηz=∑j=0∞ℍj,λkζ;uzjj!∑r=0∞ηr,λzrr!=∑j=0∞∑r=0jjrℍj−r,λkζ;uηr,λzjj!.
Thus, by Equation (38), we complete the proof.☐Theorem 10.
Letj≥0. Then, we have the following:
(39)ℍj,λkζ+1;u=∑r=0jjrℍj−r,λkζ;u1r,λ.Proof.
In Equation (16), we see that
(40)∑j=0∞ℍj,λkζ+1;u−ℍj,λkζ;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζzeλz−1=∑j=0∞∑r=0jjrℍj−r,λkζ;u1r,λzjj!−∑j=0∞ℍj,λkζ;uzjj!.
Comparing the coefficients ofzj on both sides, we get the result.☐Theorem 11.
Letj≥0. Then,
(41)ℍj,λkζ;u=∑r=0j∑q=0rjrζqS2,λr,qℍj−r,λku.Proof.
From Equation (16), we see that
(42)∑j=0∞ℍj,λkζ;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζz=Eik,λlogλ1+1−uzzeλz−ueλz−1+1ζ=Eik,λlogλ1+1−uzzeλz−u∑q=0∞ζq∑l=q∞S2,λl,qzll!=∑j=0∞∑r=0j∑q=0rjrζqS2,λr,qℍj−r,λkuzjj!.
By Equation (42). We complete the proof.☐Theorem 12.
Letj≥0. Then,
(43)ℍj,λkζ+α∣u=∑n=0j∑l=0njnulζlWu,αn,l;λℍj−n,λku.Proof.
By changingζ by ζu+α in Equation (16), we get the following:
(44)∑j=0∞ℍj,λkζ+α;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλαteλζuz=Eik,λlogλ1+1−uzzeλz−ueλαteλuz−1+1ζ=Eik,λlogλ1+1−uzzeλz−ueλαz∑l=0∞ζleλuz−1l=Eik,λlogλ1+1−uzzeλz−ueλαz∑l=0∞ulζlfraceλuz−1ll!ul=∑j=0∞ℍj,λkuzjj!∑n=0∞∑l=0nulζlWu,αn,l;λznn!=∑j=0∞∑n=0j∑l=0njnulζlWu,αn,l;λℍj−n,λkuzjj!.
Therefore, by Equations (16) and (44), we obtain the result.☐


## 3. 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:
(45)uk,λζ∣p=∑j=1∞pj1j,λζjjk.Note that, we have the following:(46)uk,λζ∣1Γ=Eik,λζis the modified degenerate polylogarithm function.It is clear that(47)limλ⟶0uk,λζ∣p=∑j=1∞limλ⟶0pi1i,λζiik=ukζ∣p=∑j=1∞piζiik,k∈ℤare called the unipoly function attached to polynomials pζ (see [20]).From Equation (47), we have the following:
(48)ukζ∣1=∑j=1∞ζjjk=Likζsee9is the ordinary polylogarithm function.By using Equations (45) and (16), we define the degenerate unipoly-Frobenius-Euler polynomials as follows:
(49)uk,λlogλ1+1−uz∣pzeλz−ueλζz=∑j=0∞ℍj,λ,pkζ;uzjj!.At the special valueζ=0, ℍj,λ,pku=ℍj,λ,pk0;u are called the degenerate unipoly-Frobenius-Euler numbers.Theorem 13.
Letj≥0. Then, we have the following:
(50)ℍj,λ,1/Γkζ;η=ℍj,λkζ;u,k∈ℤ.Proof.
Let us takepj=1/Γλ. Then, we have the following:
(51)∑j=0∞ℍj,λ,1/Γkζ;uzjj!=uk,λlogλ1+1−uz∣1/Γpzeλz−ueλζz=1zeλz−u∑r=1∞1r,λlogλ1+1−uzrrkr+1!eλζz=Eik,λlogλ1+1−uzzeλz−ueλζz=∑j=0∞ℍj,λkζ;uzjj!.
In view of Equation (51), we complete the proof.☐Theorem 14.
Letj∈ℕ and k∈ℤ. Then,
(52)ℍj,λ,pkζ;u=∑s=0j∑r=0sjspr+11r+1,λr+1!S1,λr+1,s+1ℍj−s,λu1−usr+1ks+1.Proof.
Using Equation (49), we have the following:
(53)∑j=0∞ℍj,λ,pkuzjj!=uk,λlogλ1+1−uz∣pzeλz−u=1zeλz−u∑r=1∞pr1r,λrklogλ1+1−uzr=1zeλz−u∑r=0∞pr+11r+1,λr+1!r+1k∑l=r+1∞S1,λr+1,l1−ulzll!=∑j=0∞ℍj,λuzjj!∑r=0∞∑l=0rpr+11r+1,λr+1!r+1kS1,λr+1,l+1l+1zll!=∑j=0∞∑s=0∞∑r=0sjspr+11r+1,λr+1!S1,λr+1,s+1ℍj−s,λu1−usr+1ks+1zjj!.
Therefore, by Equations (49) and (53), we get the result.☐Corollary 15.
Letj≥0. Then, we have the following:
(54)ℍj,λ,1/Γku=ℍj,λku=∑s=0n∑r=0sjlS1,λr+1,s+1ℍj−s,λur+1k−1s+1.Theorem 16.
Letj≥0. Then, we have the following:
(55)ℍj,λ,pkζ;u=∑r=0j∑s=0rjrHj−r,λ,pkuζrS2,λr,s.Proof.
By Equation (49), we have the following:
(56)∑j=0∞Hj,λk,px;uzjj!=uk,λlogλ1+1−uz∣pzeλz−ueλz−1+1ζ=uk,λlogλ1+1−uz∣pzeλz−u∑l=0∞ζleλz−1l=∑j=0∞Hj,λ,pkuzjj!∑l=0∞ζl∑m=l∞S2,λm,lzmm!=∑j=0∞∑r=0j∑s=0rnrHj−r,λ,pkuζrS2,λr,szjj!.
By Equation (56), we obtain the result.☐Theorem 17.
Letj≥0. Then,
(57)ℍj,λ,pkζ;u=jℍj−1,λ,pkζ;u.Proof.
When we consider Equation (49), we see that
(58)Δλ∑j=0∞ℍn,λ,pkζ;uzjj!=Δλuk,λlogλ1+1−uz∣pzeλz−u1+λzv/λ,and then we have the following:
(59)∑j=0∞Δλℍj,λ,pkζ;uzjj!=uk,λlogλ1+1−uzzeλz−uΔλeλζz=uk,λlogλ1+1−uzzeλz−ueλζzz=∑j=0∞ℍj,λ,pkζ;uzj+1j!.
Therefore, by Equation (59), we complete the proof.☐Theorem 18.
Letj≥0 and k∈ℤ. Then, we have the following:
(60)∂∂ζℍj,λ,pkζ;u=∑r=0jjrℍj−r,λ,pkζ;u1r,λ.Proof.
In Equation (49), we consider that
(61)∂∂ζ∑j=0∞ℍj,λ,pkζ;uzjj!=∂∂ζuk,λlogλ1+1−uz∣pzeλz−u1+λzζ/λ∑j=0∞∂∂ζℍj,λ,pkζ;uzjj!=uk,λlogλ1+1−uzzeλz−u∂∂ζ1+λzζ/λ=uk,λlogλ1+1−uzzeλz−u1+λzζ/λ1+λz1/λ=∑j=0∞ℍj,λ,pkζ;uzjj!∑r=0∞1r,λzrr!=∑j=0∞∑r=0jjrℍj−r,λ,pkζ;u1r,λzjj!.
By Equation (61), we complete the proof.☐


## 4. Degenerate Poly-Frobenius-Euler Polynomials of Complex Variables

In this section, we define the Frobenius-Euler polynomials of the complex variables. We consider the degenerate cosine function and degenerate sine function. Using the degenerate cosine function and the degenerate sine function, we introduce the degenerate cosine poly-Frobenius-Euler polynomials and degenerate sine poly-Frobenius-Euler polynomials.In [5], Kim et al. defined the degenerate sine sinλz and cosine cosλz functions by the following:
(62)sinλζz=eλixz−eλ−iζz2i,cosλζz=eλiζz+eλ−iζz2,where i=−1. Note that limλ⟶0sinλζz=sinζz and limλ⟶0cosλζz=cosζz. From Equation (62), it is readily seen that
(63)eλiζz=cosλζz+isinλζz.By these functions in Equation (62), the degenerate sine-polynomials Sj,λζ,η and degenerate cosine-polynomials Cj,λζ,η are introduced by Kim et al. [5] as follows:
(64)∑j=0∞Sj,λζ,ηzjj!=eλζzsinληz,(65)∑j=0∞Cj,λζ,ηzjj!=eλζzcosληz.Several properties of these polynomials in Equations (64) and (65) were studied and investigated by Kim et al. [5]. Also, by means of these functions, Kim et al. [5] introduced the degenerate Euler and Bernoulli polynomials of complex variable and investigate some of their properties. Motivated and inspired by these considerations above, we define degenerate poly-Frobenius-Euler polynomials of complex variable as follows:
(66)∑j=0∞ℍj,λkζ+iη;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζ+iηz=Eik,λlogλ1+1−uzzeλz−ueλζzcosληz+isinληz,(67)∑j=0∞ℍj,λkζ−iη;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζ−iηz=Eik,λlogλ1+1−uzzeλz−ueλζzcosληz−isinληz.From Equations (66) and (67), we get the following:
(68)Eik,λlogλ1+1−uzzeλz−ueλζzcosληz=∑j=0∞ℍj,λkζ+iη;u+ℍj,λkζ−iη;u2zjj!,Eik,λlogλ1+1−uzzeλz−ueλζzsinληz=∑j=0∞ℍj,λkζ+iη;u−ℍj,λkζ−iη;u2izjj!.Now, we define the degenerate cosine poly-Frobenius-Euler polynomials and the degenerate sine poly-Frobenius-Euler polynomials, respectively, as follows:(69)∑j=0∞ℍj,λk,cζ,η;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζzcosληz,(70)∑j=0∞ℍj,λk,sζ,η;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζzsinληz.Theorem 19.
The following results hold true:(71)ℍj,λk,cζ,η;u=12∑r=0jjrℍj−r,λk,cζ,η;u∑l=0rrlζr−l,λiηl,λ+−iηl,λ,(72)ℍj,λk,sζ,η;u=12i∑r=0jjrℍj−r,λk,sζ,η;u∑l=0rrlζr−l,λiηl,λ−−iηl,λ.Proof.
From Equation (69), we have the following:
(73)∑j=0∞ℍj,λk,cζ,η;uzjj!=Eik,λlogλ1+1−uzzeλz−ueλζzcosληz=Eik,λlogλ1+1−uzzeλz−u12∑r=0∞∑l=0rrlζr−l,λiηl,λ+−iηl,λzrr!=∑j=0∞12∑r=0jjrℍj−r,λk,cζ,η;u∑l=0rrlζr−l,λiηl,λ+−iηl,λzjj!.
By Equation (73), we get Equation (71). Similarly, by using Equations (70) and (3), we can easily find the result (Equation (72)).☐Theorem 20.
The following results hold true:(74)ℍj,λk,cζ1+ζ2,;u=∑m=0jjm∑r=0mS2,λζ1m,rζ2rℍj,λk,c0,η;u,(75)ℍj,λk,sζ1+ζ2,;u=∑m=0jjm∑r=0mS2,λζ1m,rζ2rℍj,λk,s0,η;u.Proof.
From Equations (3) and (69), we have the following:
(76)∑j=0∞ℍj,λk,cζ1+ζ2,;uzjj!=eλζ1+ζ2zEik,λlogλ1+1−uzzeλz−ucosληz=∑m=0∞∑r=0mS2,λζ1m,rζ2rzmm!∑j=0∞ℍj,λk,c0,η;uzjj!=∑j=0∞∑m=0jjm∑r=0mS2,λζ1m,rζ2rℍj,λk,c0,η;uzjj!.
In view of Equation (76), we get Equation (74). Similarly, by using Equations (3) and (70), we require at the desired result (Equation (75)).☐Theorem 21.
The following results hold true:(77)ℍj,λk,cζ1+ζ2,η1+η2;u=∑r=0jjrℍj−r,λk,cζ1,η1;uCr,λζ2,η2−ℍj,λk,sζ1,η1;uSr,λζ2,η2,(78)ℍj,λk,sζ1+ζ2,η1+η2;u=∑r=0jjrℍj−r,λk,sζ1,η1;uCr,λζ2,η2−ℍj,λk,cζ1,η1;uSm,λζ2,η2.Proof.
From Equation (69), we have the following:
(79)∑j=0∞ℍj,λk,cζ1+ζ2,η1+η2;uzjj!=eλζ1+ζ2zEik,λlogλ1+1−uzzeλz−ucosλη1zcosλη2z−sinλη1zsinλη2z=∑j=0∞ℍj,λk,cζ1,η1;uzjj!∑r=0∞Cr,λζ2,η2zrr!−∑j=0∞ℍj,λk,sζ1,η1;uzjj!∑r=0∞Sr,λζ2,η2zrr!=∑j=0∞∑r=0jjrℍj−r,λk,cζ1,η1;uCr,λζ2,η2−ℍj,λk,sζ1,η1;uSm,λζ2,η2zjj!.
Comparing the coefficients ofz on both sides, we get Equation (77). Similarly, by using Equation (70), we can easily get Equation (78).☐Corollary 22.
On settingζ1=ζ2=ζ and η1=η2=η in Theorem 16, we have the following:
(80)ℍj,λk,c2ζ,2η;u=∑r=0jjrℍj−r,λk,cζ,η;uCr,λζ,η−ℍj,λk,sζ,η;uSr,λζ,η,ℍj,λk,s2ζ,2η;u=∑r=0jjrℍj−r,λk,sζ,η;uCr,λζ,η−ℍj,λk,cζ,η;uSm,λζ,η.


## 5. Conclusions

In this paper, we defined the degenerate poly-Frobenius-Euler polynomials by employing the modified degenerate polyexponential functions. We have established some identities and relations between degenerate Whitney numbers and degenerate Stirling numbers of the first kind. Also, we have established addition formulas and derivative formulas of degenerate poly-Frobenius-Euler polynomials. In the last section, we have defined degenerate poly-Frobenius-Euler polynomials of complex variables and then we have derived several properties and relations.



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