# A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

Authors: Waseem A. Khan; Rifaqat Ali; Khaled Ahmad Hassan Alzobydi; Naeem Ahmed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of these polynomials systematically. Then, we also consider the degenerate unipoly-Genocchi polynomials attached to an arithmetic function, by using the degenerate polylogarithm function, and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.


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## Body

## 1. Introduction

In [1, 2], Carlitz initiated a study of degenerate versions of some special polynomials and numbers, namely, the degenerate Bernoulli and Euler polynomials and numbers. Kim et al. [3–5] have studied the degenerate versions of special numbers and polynomials actively. These ideas provide a powerful tool in order to define special numbers and polynomials of their degenerate versions. The notion of degenerate version forms a special class of polynomials because of their great applicability. Despite the applicability of special functions in classical analysis and statistics, they also arise in communication systems, quantum mechanics, nonlinear wave propagation, electric circuit theory, electromagnetic theory, etc. In particular, Genocchi numbers have been extensively studied in many different contexts in such branches of mathematics as, for instance, elementary number theory, complex analytic number theory, differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), p-adic analytic number theory (p-adic L-functions), and quantum physics (quantum groups). The works of Genocchi numbers and their combinatorial relations have received much attention [6–11]. In the paper, we focus on a new type of degenerate poly-Genocchi polynomial and numbers.The aim of this paper is to introduce a degenerate version of the poly-Genocchi polynomials and numbers, the so-called new type of degenerate poly-Genocchi polynomials and numbers, constructing from the degenerate polylogarithm function. We derive some explicit expressions and identities for those numbers and polynomials.The classical Euler polynomialsEnx and the classical Genocchi polynomials Gnx are, respectively, defined by the following generating functions (see [12–22]):
(1)2et+1ext=∑n=0∞Enxtnn!,t<π,(2)2tet+1ext=∑n=0∞Gnxtnn!,t<π.In the case whenx=0, En0≔En and Gn0≔Gn are, respectively, called the Euler numbers and Genocchi numbers.The degenerate exponential function [23, 24] is defined by
(3)eλxt=1+λtx/λ,eλ1t=eλtλ∈ℝ.Note that(4)limλ⟶01+λtx/λ=∑n=0∞xntnn!=ext.In [1, 2], Carlitz introduced the degenerate Bernoulli and degenerate Euler polynomials defined by
(5)teλt−1eλxt=t1+λt1/λ−11+λtx/λ=∑n=0∞βnx;λtnn!,(6)2eλt+1eλxt=21+λt1/λ−11+λtx/λ=∑n=0∞Enx;λtnn!.In the case whenx=0, Bn,λ0≔Bn,λ are called the degenerate Bernoulli numbers and En,λ0≔En,λ are called the degenerate Euler numbers.Letxn,λ be the degenerate falling factorial sequence given by
(7)xn,λ≔xx−λ⋯x−n−1λn≥1,with the assumption x0,λ=1.In [5], Kim et al. considered the degenerate Genocchi polynomials given by
(8)2teλt+1eλxt=∑n=0∞Gn,λxtnn!.In the case whenx=0, Gn,λ≔Gn,λ0 are called the degenerate Genocchi numbers.Fork∈ℤ, the polylogarithm function is defined by a power series in t, which is also a Dirichlet series in k (see [25, 26]):
(9)Likt=∑n=1∞tnnk=t+t22k+t33k+⋯t<1.This definition is valid for arbitrary complex orderk and for all complex arguments t with t<1: it can be extended to t≥1 by analytic continuation.It is noticed that(10)Li1t=∑n=1∞tnn=−log1−t.Forλ∈ℝ, Kim and Kim [3] defined the degenerate version of the logarithm function, denoted by logλ1+t, as follows (see [4]):
(11)logλ1+t=∑n=1∞λn−11n,1/λtnn!,being the inverse of the degenerate version of the exponential function eλt as has been shown below:
(12)eλlogλt=logλeλt=t.It is noteworthy to mention that(13)limλ⟶0logλ1+t=∑n=1∞−1n−1tnn!=log1+t.The degenerate polylogarithm function [3] is defined by Kim and Kim to be
(14)lk,λx=∑n=1∞−λn−11n,1/λn−1!nkxnk∈ℤ,x<1.It is clear that (see [27, 28])
(15)limλ⟶0lk,λx=∑n=1∞xnnk=Likx.From (11) and (14), we get
(16)l1,λx=∑n=1∞−λn−11n,1/λn!xn=−logλ1−x.Very recently, Kim and Kim [3] introduced the new type of degenerate version of the Bernoulli polynomials and numbers, by using the degenerate polylogarithm function as follows:
(17)lk,λ1−eλ−t1−eλ−teλxt=∑n=0∞βn,λkxtnn!.Whenx=0, βj,λk≔βj,λk0 are called the new type of degenerate poly-Bernoulli numbers.The degenerate Stirling numbers of the first kind [24] are defined by
(18)1k!logλ1+tk=∑n=k∞S1,λn,ktnn!k≥0.It is clear that(19)limλ⟶0S1,λn,k≔S1n,k,calling the Stirling numbers of the first kind given by (see [29, 30])
(20)1k!log1+tk=∑n=k∞S1n,ktnn!k≥0.The degenerate Stirling numbers of the second kind [31] are given by (see [2, 13–22, 25–32])
(21)1k!eλt−1k=∑n=k∞S2,λj,ktnn!k≥0.Note here that(22)limλ⟶0S2,λn,k≔S2n,k,standing for the Stirling numbers of the second kind given by means of the following generating function (see [1–8, 12–38]):
(23)1k!et−1k=∑n=k∞S2n,ktnn!k≥0.This paper is organized as follows. In Section1, we recall some necessary stuffs that are needed throughout this paper. These include the degenerate exponential functions, the degenerate Genocchi polynomials, the degenerate Euler polynomials, and the degenerate Stirling numbers of the first and second kinds. In Section 2, we introduce the new type of degenerate poly-Genocchi polynomials by making use of the degenerate polylogarithm function. We express those polynomials in terms of the degenerate Genocchi polynomials and the degenerate Stirling numbers of the first kind and also of the degenerate Euler polynomials and the Stirling numbers of the first kind. We represent the generating function of the degenerate poly-Genocchi numbers by iterated integrals from which we obtain an expression of those numbers in terms of the degenerate Bernoulli numbers of the second kind. In Section 3, we introduce the new type of degenerate unipoly-Genocchi polynomials by making use of the degenerate polylogarithm function. We express those polynomials in terms of the degenerate Genocchi polynomials and the degenerate Stirling numbers of the first kind and also of the degenerate Euler polynomials and the Stirling numbers of the first kind and second kind.


## 2. New Type of Degenerate Genocchi Numbers and Polynomials

In this section, we define the new type of degenerate Genocchi numbers and polynomials by using the degenerate polylogarithm function which is called the degenerate poly-Genocchi polynomials as follows.Fork∈ℤ, we define the new type of degenerate Genocchi numbers, which are called the degenerate poly-Genocchi numbers, as
(24)2eλt+1lk,λ1−eλ−t=∑n=0∞Gn,λktnn!.Note that(25)∑n=0∞Gn,λ1tnn!=2eλt+1l1,λ1−eλ−t=2teλt+1=∑n=0∞Gn,λtnn!.Thus, we have (see [6])
(26)Gn,λ1=Gn,λn≥0.Now, we consider the new type of degenerate Genocchi polynomials which are called the degenerate poly-Genocchi polynomials defined by(27)2lk,λ1−eλ−teλt+1eλxt=∑n=0∞Gn,λkxtnn!.In the case whenx=0, Gn,λk≔Gn,λk0. Using equation (27), we see
(28)∑n=0∞Gn,λkxtnn!=2lk,λ1−eλ−teλt+1eλxt=∑m=0∞Gm,λktmm!∑n=0∞xn,λtnn!=∑n=0∞∑m=0nn/mGm,λkxn−m,λtnn!.Therefore, by equation (28), we obtain the following theorem.Theorem 1.
Letn be a nonnegative integer. Then,
(29)Gn,λkx=∑m=0nn/mGm,λkxn−m,λ.
From (27), we note that
(30)∑n=0∞Gn,λkxtnn!=2teλt+1eλxt1tlk,λ1−eλ−t,(31)∑n=0∞Gn,λkxtnn!=2teλt+1eλxt1t∑m=1∞−λm−11m,1/λm−1!mk1−eλ−tm=2teλt+1eλxt1t∑m=1∞−λm−11m,1/λmk−1∑l=m∞−1l−mS2,λl,mtll!=2teλt+1eλxt1t∑l=1∞∑m=1lλm−11m,1/λ−1l−1mk−1S2,λl,mtll!=∑n=0∞Gn,λxtnn!1t∑l=0∞∑m=1l+1λm−11m,1/λ−1lmk−1S2,λl+1,mtl+1l+1!=∑n=0∞Gn,λxtnn!∑l=0∞∑m=1l+1λm−11m,1/λ−1lmk−1S2,λl+1,ml+1tll!=∑n=0∞∑l=0nnl∑m=1l+1λm−11m,1/λ−1lmk−1S2,λl+1,ml+1Gn−l,λxtnn!.Therefore, by equations (30) and (31), we get the following theorem.Theorem 2.
Letn be a nonnegative integer. Then,
(32)Gn,λkx=∑l=0nnl∑m=1l+1λm−11m,1/λ−1lmk−1S2,λl+1,ml+1Gn−l,λx.
Using equations (27) and (6), we see
(33)∑n=0∞Gn,λkxtnn!=2eλt+1eλxtlk,λ1−eλ−t,(34)∑n=0∞Gn,λkxtnn!=2eλt+1eλxt∑m=1∞−λm−11m,1/λm−1!mk1−eλ−tm=2eλt+1eλxt∑m=1∞−λm−11m,1/λmk−1∑l=m∞−1l−mS2,λl,mtll!=2eλt+1eλxt∑l=1∞∑m=1lλm−11m,1/λ−1l−1mk−1S2,λl,mtll!=∑n=0∞En,λxtnn!∑l=1∞∑m=1lλm−11m,1/λ−1l−1mk−1S2,λl,mtll!=∑n=1∞∑l=1nnl∑m=1lλm−11m,1/λ−1l−1mk−1S2,λl,mEn−l,λxtnn!.By equations (33) and (34), we obtain the following theorem.Theorem 3.
Letn be a nonnegative integer. Then,
(35)Gn,λkx=∑l=1nnl∑m=1lλm−11m,1/λ−1l−1mk−1S2,λl,mEn−l,λx.
From (27), we have
(36)∑n=0∞Gn,λkxnn!=2eλx+1lk,λ1−eλ−x=2eλx+1∫0xeλ1−λ−t1−eλ−t∫0teλ1−λ−t1−eλ−t⋯∫0teλ1−λ−t1−eλ−t⏟k−2−timestdtdt⋯dt∑n=0∞Gn,λkxnn!=2eλx+1∫0xeλ1−λ−t1−eλ−t∫0teλ1−λ−t1−eλ−t⋯∫0teλ1−λ−t1−eλ−ttdtdt⋯dt⏟k−2−times.
Fork=2 in (36) and using [3] (Eq. (27)), we get
(37)∑n=0∞Gn,λ2xnn!=2eλx+1∫0xt1−eλ−teλ1−λ−tdt=2eλx+1∫0x∑j=0∞βj,λ1−λ−1jtjj!dt=2xeλx+1∑j=0∞βj,λ1−λj+1−1jxjj!=∑n=0∞Gn,λxnn!∑j=0∞βj,λ1−λj+1−1jxjj!=∑n=0∞∑j=0nn/j−1jGn−j,λβj,λ1−λj+1xnn!.Therefore, by equation (37), we get the following theorem.Theorem 4.
Letn be a nonnegative integer. Then,
(38)Gn,λkx=∑m=0nn/mGm,λkxn−m,λ.
In general, by equation (37), we see
(39)∑n=0∞Gn,λkxnn!=2eλx+1∫0xeλ1−λ−t1−eλ−t∫0teλ1−λ−t1−eλ−t⋯∫0teλ1−λ−t1−eλ−ttdtdt⋯dt=∑n1,n2,⋯,nk−1=n∞1n1!n2!⋯nk−1!βn1,λ1−λn1+1βn2,λ1−λn1+n2+1×⋯βnk−1,λ1−λn1+⋯+nk−1+1−xn1,n2,⋯,nk−12xeλx+1=∑n=0∞−1n∑n1,n2,⋯,nk=nnn1,n2,⋯,nkβn1,λ1−λn1+1βn2,λ1−λn1+n2+1⋯βnk−1,λ1−λn1+⋯+nk−1+1Gn,λxnn!.By equation (39), we obtain the following theorem.Theorem 5.
Letk∈ℤ and n≥0, we have
(40)Gn,λk=−1n∑n1,n2,⋯,nk=nnn1,n2,⋯,nkβn1,λ1−λn1+1βn2,λ1−λn1+n2+1⋯βnk−1,λ1−λn1+⋯+nk−1+1Gn,λ.
From (27), we observe that
(41)2lk,l1−eλ−t=1+eλt∑m=0∞Gm,λktmm!=∑j=1∞Gj,λk+Gj,λk1tjj!.
On the other hand,(42)2lk,l1−eλ−t=2∑r=1∞−λr−11r,1/λr−1!rk1−eλ−tr=2∑m=1∞−λm−11m,1/λmk−11m!1−eλ−tm=2∑r=1∞−λr−11r,1/λrk−1∑j=r∞S2,λj,r−1j−rtjj!=2∑j=1∞∑r=1j−1j−11r,1/λrk−1λr−1S2,λj,rtjj!.Therefore, by equations (41) and (42), we get the following theorem.Theorem 6.
Letk∈ℤ and j≥1. Then,
(43)12Gj,λk+Gj,λk1=−1j−1∑r=1j1r,1/λrk−1λr−1S2,λj,r.
From equations (27) and (14), we see
(44)2t=2l1,l1−eλ−t=2∑m=1∞−λm−11m,1/λm−1!mk1−eλ−tm=2∑m=1∞−λm−11m,1/λm!1−eλ−tm=2∑m=1∞−λm−11m,1/λ∑n=m∞S2,λn,m−1n−mtnn!=2∑n=1∞∑m=1n−1n−11m,1/λλm−1S2,λn,mtnn!.By comparing the coefficients on both sides of (44), we obtain the following theorem.Theorem 7.
Forn∈ℕ, we have
(45)∑m=1n−1n−11m,1λλm−1S2,λn,m=δn,1,where δn,k is Kronecker’s symbol.
Note that(46)limλ⟶0Gn,λ1=Gn,limλ⟶0Gn,λ1x=Gnx.


## 3. Degenerate Unipoly-Genocchi Numbers and Polynomials

Letp be any arithmetic function which is a real or complex valued function defined on the set of positive integers ℕ. Kim and Kim [29] defined the unipoly function attached to polynomials px by
(47)ukx∣p=∑n=1∞pnnkxnk∈ℤ.Moreover (see [25]),
(48)ukx∣1=∑n=1∞xnnk=Likxis the ordinary polylogarithm function.In [8], Lee and Kim defined the degenerate unipoly function attached to polynomials px as follows:
(49)uk,λx∣p=∑i=1∞pi−λi−11i,1/λikxi.It is worthy to note that(50)uk,λx∣1Γ=lk,λxis the degenerate polylogarithm function.Now, we define the degenerate unipoly-Genocchi polynomials attached to polynomialspx by
(51)2uk,λ1−eλ−t∣peλt+1eλxt=∑n=0∞Gn,λ,pkxtnn!.In the case whenx=0, Gn,λ,pk≔Gn,λ,pk0 are called the degenerate unipoly-Genocchi numbers attached to p.From (51), we see
(52)∑n=0∞Gn,λ,1/Γktnn!=2eλt+1uk,λ1−eλ−t∣1Γ=2eλt+1∑r=1∞−λr−11r,1/λ1−eλ−trrkr−1!=2eλt+1lk,λ1−eλ−t=∑n=0∞Gn,λktnn!.Thus, by (52), we have
(53)Gn,λ,1Γk=Gn,λk.From (51), we have
(54)∑n=0∞Gn,λ,pkxtnn!=2eλxteλt+1uk,λ1−eλ−t∣p=2eλxteλt+11t∑m=1∞pm−λm−11m,1/λmk1−eλ−tm=2teλt+1eλxt1t∑m=1∞pm−λm−11m,1/λm!mk∑l=m∞−1l−mS2,λl,mtll!=2teλt+1eλxt1t∑l=1∞∑m=1lλm−11m,1/λ−1l−1m!mkS2,λl,mtll!=∑n=0∞Gn,λxtnn!1t∑l=0∞∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,mtl+1l+1!=∑n=0∞Gn,λxtnn!∑l=0∞∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1tll!=∑n=0∞∑l=0nnl∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1Gn−l,λxtnn!.Therefore, by equation (54), we get the following theorem.Theorem 8.
Letn be a nonnegative integer. Then,
(55)Gn,λ,pkx=∑l=0nnl∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1Gn−l,λx.
Using equations (49) and (51), we see
(56)∑n=0∞Gn,λkxtnn!=2eλt+1eλxt∑m=1∞pm−λm−11m,1/λmk1−eλ−tm=2eλt+1eλxt∑m=1∞pm−λm−11m,1/λm!mk∑l=m∞−1l−mS2,λl,mtll!=2eλt+1eλxt∑l=1∞∑m=1lpmλm−11m,1/λ−1l−1m!mkS2,λl,mtll!=∑n=0∞En,λxtnn!∑l=1∞∑m=1lpmλm−11m,1/λ−1l−1m!mkS2,λl,mtll!=∑n=1∞∑l=1njl∑m=1lpmλm−11m,1/λ−1l−1m!mkS2,λl,mEn−l,λxtnn!.By, equations (51) and (56), we obtain the following theorem.Theorem 9.
Letn be a nonnegative integer. Then,
(57)Gn,λ,pkx=∑l=1nnl∑m=1lpmλm−11m,1/λ−1l−1m!mkS2,λl,mEn−l,λx.
From (6), (49), and (51), we get
(58)∑n=0∞Gn,λ,pkxtnn!=2eλxteλt+1uk,λ1−eλ−t∣p=2teλt+1eλt−1eλt−1eλxt1t∑m=1∞pm−λm−11m,1/λmk1−eλ−tm=2teλxteλ2t−1eλt−11t∑m=1∞pm−λm−11m,1/λmk1−eλ−tm=2teλ/2x/22teλ/22t−1eλt−1∑l=0∞∑m=1l+1pm−1lλm−11m,1/λm!mkS2,λl+1,mtll+1!=∑n=0∞βn,λ/2x22ntnn!∑i=1∞1i,λtii!∑l=0∞∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1tll!=∑n=0∞βn,λ/2x22ntnn!∑i=0∞1i+1,λi+1tii!∑l=0∞∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1tll!∑n=0∞βn,λ/2x22ntnn!∑i=0∞∑l=0iil1i−l+1,λi−l+1∑m=1l+1pmλm−11m,1/λ−1lm!mkS2,λl+1,ml+1tii!=∑n=0∞∑i=0n∑l=0iniil∑m=1l+11i−l+1,λpmλm−11m,1/λ−1lm!i−l+1mkS2,λl+1,ml+12n−iβn−i,λ/2x2tnn!.Therefore, by (58), we obtain the following theorem.Theorem 10.
Letn be a nonnegative integer and k∈ℤ. Then,
(59)Gn,λ,pkx=∑i=0n∑l=0iniil∑m=1l+11i−l+1,λpmλm−11m,1/λ−1lm!i−l+1mkS2,λl+1,ml+1×2n−iβn−i,λ/2x2.
From (51), we have
(60)∑n=0∞Gn,λ,pkxtnn!=2eλt+1uk,λ1−eλ−t∣peλt−1+1x=2uk,λ1−eλ−t∣peλt+1∑i=0∞xieλt−1ii!=∑n=0∞Gn,λ,pktnn!∑i=0∞xi∑l=i∞S2,λl,itll!=∑n=0∞Gn,λ,pktnn!∑i=0∞∑i=0lxiS2,λl,itll!=∑n=0∞∑l=0n∑i=0lnlxiS2,λl,iGn−l,λ,pktnn!.By equation (60), we get the following theorem.Theorem 11.
Letn be a nonnegative integer and k∈ℤ. Then,
(61)Gn,λ,pkx=∑l=0n∑i=0lnlxiS2,λl,iGn−l,λ,pk.


## 4. Conclusion

In this article, we introduced degenerate poly-Genocchi polynomials and numbers by using the degenerate polylogarithm function and derived several properties on the degenerate poly-Genocchi numbers. We represented the generating function of the degenerate poly-Genocchi numbers by iterated integrals in Theorems4–6 and explicit degenerate poly-Genocchi polynomials in terms of the Euler polynomials and degenerate Stirling numbers of the second kind in Theorem 3. We also represented those numbers in terms of the degenerate Stirling numbers of the second kind in Theorem 7. In the last section, we defined the degenerate unipoly-Genocchi polynomials by using degenerate polylogarithm function and obtained the identity degenerate unipoly-Genocchi polynomials in terms of the degenerate Genocchi polynomials and degenerate Stirling numbers of the second kind in Theorem 8, the degenerate Euler polynomials and the degenerate Stirling numbers of the second kind in Theorem 9, the degenerate Bernoulli and degenerate Stirling numbers of the second kind in Theorem 10, and the degenerate unipoly-Genocchi numbers and Stirling numbers of the second kind in Theorem 11. It is important that the study of the degenerate version is widely applied not only to numerical theory and combinatorial theory but also to symmetric identity, differential equations, and probability theory. In particular, many symmetric identities have been studied for degenerate versions of many special polynomials [1, 3, 12, 23, 29–32]. Genocchi numbers have been also extensively studied in many different branches of mathematics. The works of Genocchi numbers and their combinatorial relations have received much attention [6–9]. With this in mind, as a future project, we would like to continue to study degenerate versions of certain special polynomials and numbers and their applications to physics, economics, and engineering as well as mathematics.



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