# Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

Authors: G. Muhiuddin; W. A. Khan; U. Duran; D. Al-Kadi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.


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## Body

## 1. Introduction

For anyλ∈ℝ/0 (or ℂ/0), degenerate version of the exponential function eλxt is defined as follows (see [1–15])
(1)eλxt≔1+λtxλ=∑n=0∞xn,λtnn!,where x0,λ=1 and xn,λ=xx−λ⋯x−n−1λ for n≥1, (cf. [1–15]). It follows from (1) is limλ→0eλxt=ext. Note that eλ1t≔eλt.Carlitz [1] introduced the degenerate Bernoulli polynomials as follows:
(2)teλt−1eλxt=∑n=0∞βnx;λtnn!.Upon settingx=0, βn0;λ≔βnλ are called the degenerate Bernoulli numbers.Note that(3)limλ→0βnx;λ=Bnx,where Bnx are the familiar Bernoulli polynomials (cf. [1, 3, 4, 6, 8, 11, 12, 14, 16–22])
(4)tet−1ext=∑n=0∞Bnxtnn!,t<2π.Fork∈ℤ, the polyexponential function Eikx is defined by (see [21])
(5)Eikx=∑n=1∞xnn−1!nk,k∈ℤ.Setting k=1 in (5), we have Ei1x=ex−1.The degenerate modified polyexponential function [12] is defined, for k∈ℤ and x<1, by
(6)Eik;λx=∑n=1∞1n,λn−1!nkxn.Note that Ei1;λx=eλx−1.Letk∈ℤ and λ∈ℝ. The degenerate poly-Bernoulli polynomials, cf. [12], are defined by
(7)Eik;λlogλ1+teλt−1eλxt=∑n=0∞Bn,λkxtnn!,(8)logλ1+t=∑n=1∞λn−11n,1λtnn!,λ∈ℝ,where logλ1+t are called the degenerate version of the logarithm function (cf. [8, 12]), which is also the inverse function of the degenerate exponential function eλt as shown below (cf. [8])
(9)eλlogλ1+t=logλeλ1+t=1+t.Letting x=0 in (7), Bn,λk0≔Bn,λk are called the type 2 degenerate poly-Bernoulli numbers.The degenerate Stirling numbers of the first kind (cf. [8, 13]) and second kind (cf. [4–6, 9, 17]) are defined, respectively, by
(10)1k!logλ1+tk=∑n=k∞S1,λn,ktnn!,k≥0,and (cf. [1–27])
(11)1k!eλt−1k=∑n=k∞S2,λn,ktnn!,k≥0.Note that limλ→0 in (10) and (1.8), we have (cf. [8, 13])
(12)log1+tkk!=∑n=k∞S1n,ktnn!k≥0,and (cf. [4–6, 9, 17, 24])
(13)et−1kk!=∑n=k∞S2n,ktnn!k≥0,where S1n,k and S2n,k are called the Stirling numbers of the first kind and second kind.The following paper is as follows. In Section2, we define the degenerate multi-poly-Bernoulli polynomials and numbers by using the degenerate multiple polyexponential functions and derive some properties and relations of these polynomials. In Section 3, we consider the degenerate multi-poly-Bernoulli polynomials of a complex variable and then we derive several properties and relations. Also, we examine the results derived in this study [28, 29].


## 2. Degenerate Multi-Poly-Bernoulli Polynomials and Numbers

Letk1,k2,⋯,kr∈ℤ. The degenerate multiple polyexponential function Eik1,k2,⋯,kr;λx is defined (cf. [15]) by
(14)Eik1,k2,⋯,kr;λx=∑0<n1<n2<⋯<nr1n1,λ⋯1nr,λxnrn1−1!⋯nr−1!n1k1⋯nrkr,where the sum is over all integers n1,n2,⋯,nr satisfying 0<n1<n2<⋯<nr. Utilizing this function, Kim et al. [15] introduced and studied the degenerate multi-poly-Genocchi polynomials given by
(15)2rEik1,k2,⋯,kr;λlogλ1+teλt+1reλxt=∑n=0∞gn,λk1,k2,⋯krxtnn!.Inspired by the definition of degenerate multi-poly-Genocchi polynomials, using the degenerate multiple polyexponential function (14), we give the following definition.Definition 1.
Letk1,k2,⋯,kr∈ℤ and λ∈ℝ, we consider the degenerate multi-poly-Bernoulli polynomials are given by
(16)r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxt=∑n=0∞Bn,λk1,k2,⋯,krxtnn!.Upon settingx=0 in (16), the degenerate multi-poly-Bernoulli polynomials reduce to the corresponding numbers, namely, the type 2 degenerate multi-poly-Bernoulli numbers Bn,λk1,k2,⋯,kr0≔Bn,λk1,k2,⋯,kr.Remark 2.
Asλ→0, the degenerate multi-poly-Bernoulli polynomials reduce to the multi-poly-Bernoulli polynomials given by
(17)r!Eik1,k2,⋯,krlog1+tet−1rext=∑n=0∞Bnk1,k2,⋯,krxtnn!.Remark 3.
Upon settingr=1 in (16), the degenerate multi-poly-Bernoulli polynomials reduce to the degenerate poly-Bernoulli polynomials in (7).Before going to investigate the properties of the degenerate multi-poly-Bernoulli polynomials, we first give the following result.Proposition 4 (Derivative Property).
Fork1,k2,⋯,kr∈ℤ and λ∈ℝ, we have
(18)ddxEik1,k2,⋯,kr;λx=1xEik1,k2,⋯,kr−1;λx.Proof.
By (14), we see that
(19)ddxEik1,k2,⋯,kr;λx=ddx∑0<n1<n2<⋯<nr1n1,λ⋯1nr,λxnrn1−1!⋯nr−1!n1k1⋯nrkr=1x∑0<n1<n2<⋯<nr1n1,λ⋯1nr,λxnrn1−1!⋯nr−1!n1k1⋯nrkr−1=1xEik1,k2,⋯,kr−1;λx.Theorem 5.
The following relationship(20)Bn,λk1,k2,⋯,krx=∑j=0nnjBn−j,λk1,k2,⋯krxj,λ,holds for n≥0.Proof.
Recall Definition 1 that(21)∑n=0∞Bn,λk1,k2,⋯,krxtnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxt=∑n=0∞Bn,λk1,k2,⋯,krtnn!∑j=0∞xj,λtmm!=∑n=0∞∑j=0nnjBn−j,λk1,k2,⋯krxj,λtnn!,which gives the asserted result (20).The degenerate Bernoulli polynomials of orderr are given by the following series expansion:
(22)∑n=0∞βnrx;λtnn!=teλt−1reλxt,(cf. [3, 6, 8, 17]).We provide the following theorem.Theorem 6.
Forn≥r. Then
(23)Bn,λk1,k2,⋯,krx=∑m=0n+r∑0<n1<n2<⋯<nr≤mn+rmβn+r−mrx;λS1.λm,nr×n!r!nr!1n1,λ⋯1nr,λn+r!n1−1!⋯nr−1!n1k1⋯nrkr.Proof.
Recall from Definition1 and (10) that
(24)∑n=0∞Bn,λk1,k2,⋯,krxtnn!=r!eλxteλt−1r∑0<n1<n2<⋯<nr1n1,λ⋯1nr,λlogλ1+tnrn1−1!⋯nr−1!n1k1⋯nrkr=r!eλxteλt−1r∑0<n1<n2<⋯<nr1n1,λ⋯1nr,λnr!n1−1!⋯nr−1!n1k1⋯nrkr∑m=nr∞S1.λm,nrtmm!=r!trtreλxteλt−1r∑m=nr∞∑0<n1<n2<⋯<nr≤m1n1,λ⋯1nr,λS1.λm,nrnr!n1−1!⋯nr−1!n1k1⋯nrkrtmm!=r!tr∑l=0∞βlrx;λtll!∑m=nr∞∑0<n1<n2<⋯<nr≤m1n1,λ⋯1nr,λS1.λm,nrnr!n1−1!⋯nr−1!n1k1⋯nrkrtmm!=∑n=r∞∑m=0nnm∑0<n1<n2<⋯<nr≤mr!nr!1n1,λ⋯1nr,λn1−1!⋯nr−1!n1k1⋯nrkrβn−mrx;λS1.λm,nrtn−rn!,which means the claimed result (23).Theorem 7.
The following formula(25)Bn,λk1,k2,⋯,krx+y=∑j=0nnjyj,λBn−j,λk1,k2,⋯,krx,is valid for k1,k2,⋯,kr∈ℤ and n≥0.Proof.
In view of Definition1, we see that
(26)∑n=0∞Bn,λk1,k2,⋯,krx+ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλx+yt=∑i=0∞Bi,λk1,k2,⋯,krxtii!∑j=0∞yj,λtmm!=∑n=0∞∑j=0nnjym,λBn−j,λk1,k2,⋯,krxtnn!,which implies the desired result (25).Theorem 8.
The following relation(27)ddxBn,λk1,k2,⋯,krx=l=1nnlBn−l,λk1,k2,⋯,krx−λl−1l−1!,is valid for k1,k2,⋯,kr∈ℤ and n≥0.Proof.
To investigate the derivative property ofBn,λk1,k2,⋯,krx that
(28)∑n=0∞ddxBn,λk1,k2,⋯,krxtnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1rddxeλxt=∑n=0∞Bn,λk1,k2,⋯,krxtnn!1λln1+λt=∑n=0∞Bn,λk1,k2,⋯,krxtnn!∑l=1∞−1l+1lλl−1tl=∑n=0∞∑l=1∞Bn,λk1,k2,⋯,krx−1l+1lλl−1tn+ln!,which provides the asserted result (27).We here give a relation including the degenerate multi-poly-Bernoulli polynomials with numbers and the degenerate Stirling numbers of the second kind.Theorem 9.
The following correlation(29)Bn,λk1,k2,⋯,krx=∑m=0n∑l=0mnmxlS2,λm,lBn−m,λk1,k2,⋯,kr,is valid for k1,k2,⋯,kr∈ℤ and n≥0.Proof.
By means of Definition1, we attain that
(30)∑n=0∞Bn,λk1,k2,⋯,krxtnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxt=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλt−1+1x=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1r∑l=0∞xleλt−1l=∑n=0∞Bn,λk1,k2,⋯,krtnn!∑l=0∞xl∑m=l∞S2,λm,ltmm!=∑n=0∞∑m=0n∑l=0mnmxlS2,λm,lBn−m,λk1,k2,⋯,krtnn!,where the notation xl is falling factorial that is defined by x0=1 and xn=xx−1⋯x−n−1 for n≥1, (cf. [1, 2, 5–14, 21, 23, 24]). So, the proof is completed.Kim [5] introduced the degenerate Whitney numbers are given by
(31)eλmt−1kmkk!eλαt=∑n=k∞Wm,αn,kλtnn!,k≥0.Kim also provided several correlations including the degenerate Stirling numbers of the second kind and the degenerate Whitney numbers (see [5]).We now give a correlation as follows.Theorem 10.
Fork1,k2,⋯kr∈ℤ and n≥0, we have
(32)Bn,λk1,k2,⋯,krxu+α=∑m=0n∑l=0mnmulxlWu,αm,lλBn−m,λk1,k2,⋯,kr.Proof.
Using (31) and Definition 1, we acquire that
(33)∑n=0∞Bn,λk1,k2,⋯,krxu+αtnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλαteλxut=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλαteλut−1+1x=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλαt∑l=0∞xleλut−1l=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1r∑l=0∞ulxleλut−1ll!uleλαt=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1r∑l=0∞ulxleλut−1ll!uleλαt=∑n=0∞Bn,λk1,k2,⋯,krtnn!∑n=0∞∑l=0nulxlWu,αn,lλtnn!=∑n=0∞∑m=0n∑l=0mnmulxlWu,αm,lλBn−m,λk1,k2,⋯,krtnn!,which implies the asserted result (32).


## 3. Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

In [25], Kim et al. defined the degenerate sine sinλt and cosine cosλt functions by
(34)sinλxt=eλixt−eλ−ixt2iandcosλxt=eλixt+eλ−ixt2,where i=−1. Note that limλ→0sinλxt=sinxt and limλ→0cosλxt=cosxt. From (34), it is readily that
(35)eλixt=cosλxt+isinλxt.By these functions in (34), the degenerate sine-polynomials Sk,λx,y and degenerate cosine-polynomials Ck,λx,y are introduced (cf. [25]) by
(36)∑n=0∞Sk,λx,ytnn!=eλxtsinλyt,(37)∑n=0∞Ck,λx,ytnn!=eλxtcosλyt.Several properties of these polynomials in (36) and (37) are studied and investigated in [25]. Also, by means of these functions, Kim et al. [25] 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 type 2 degenerate multi-poly-Bernoulli polynomials of complex variable as follows.Definition 11.
Letk1,k2,⋯,kr∈ℤ. We define a new form of the degenerate multi-poly-Bernoulli polynomials of complex variable by the following generating function:
(38)r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλx+iyt=∑n=0∞Bn,λk1,k2,⋯,krx+iytnn!.By (34) and (38), we observe that
(39)∑n=0∞Bn,λk1,k2,⋯,krx+iy−Bn,λk1,k2,⋯,krx−iy2itnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtsinλyt,and
(40)∑n=0∞Bn,λk1,k2,⋯,krx+iy+Bn,λk1,k2,⋯,krx−iy2tnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtcosλyt.In view of (39) and (40), we consider the degenerate multi-poly-sine-Bernoulli polynomials Bn,λk1,k2,⋯,kr;Sx,y with two parameters and the degenerate multi-poly-cosine-Bernoulli polynomials Bn,λk1,k2,⋯,kr;Cx,y with two parameters as follows:
(41)∑n=0∞Bn,λk1,k2,⋯,kr;Sx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtsinλyt,(42)∑n=0∞Bn,λk1,k2,⋯,kr;Cx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtcosλyt.Note that(43)limλ→0Bn,λk1,k2,⋯,kr;Sx,y≔Bnk1,k2,⋯,kr;Sx,yandlimλ→0Bn,λk1,k2,⋯,kr;Cx,y≔Bnk1,k2,⋯,kr;Cx,y,which are multi-poly-sine-Bernoulli polynomials Bnk1,k2,⋯,kr;Sx,y and multi-poly-cosine-Bernoulli polynomials Bnk1,k2,⋯,kr;Cx,y with two parameters.By (39)-(42), we see that
(44)Bn,λk1,k2,⋯,kr;Sx,y=Bn,λk1,k2,⋯,krx+iy−Bn,λk1,k2,⋯,krx−iy2i,Bn,λk1,k2,⋯,kr;Cx,y=Bn,λk1,k2,⋯,krx+iy+Bn,λk1,k2,⋯,krx−iy2.We now give the two summation formulae by the following theorem.Theorem 12.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(45)Bn,λk1,k2,⋯,krx+iy=∑m=0nnmBn−m,λk1,k2,⋯krxiym,λ,(46)Bn,λk1,k2,⋯,krx+iy=∑m=0nnmBn−m,λk1,k2,⋯krx+iym,λ.Proof.
The proofs of this theorem can be done by using the same proof methods used in Theorems5 and 7. So, we omit the proofs.We here provide the two derivative formulae by the following theorem.Theorem 13.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(47)ddtBn,λk1,k2,⋯,krx+iy=nBn−1,λk1,k2,⋯,krx+iy,ddxBn,λk1,k2,⋯,krx+iy=l=1nnlBn−l,λk1,k2,⋯,krx+iy−λl−1l−1!.Proof.
The proofs of this theorem can be done by using the same proof methods used in Theorem8. So, we omit the proofs.We give the following theorem.Theorem 14.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(48)Bn,λk1,k2,⋯,kr;Sx,y=l=0nnlBn−l,λk1,k2,⋯,krSl,λx,y,Bn,λk1,k2,⋯,kr;Cx,y=l=0nnlBn−l,λk1,k2,⋯,krCl,λx,y.Proof.
From (36), (37), and (38), we get
(49)∑n=0∞Bn,λk1,k2,⋯,kr;Sx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtsinλyt=∑n=0∞Bn,λk1,k2,⋯,krtnn!∑n=0∞Sn,λx,ytnn!=n=0∞l=0nnlBn−l,λk1,k2,⋯,krSl,λx,ytnn!,∑n=0∞Bn,λk1,k2,⋯,kr;Cx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtcosλyt=∑n=0∞Bn,λk1,k2,⋯,krtnn!∑n=0∞Cn,λx,ytnn!=n=0∞l=0nnlBn−l,λk1,k2,⋯,krCl,λx,ytnn!,which complete the proof of the theorem.We note that (cf. [25])
(50)sinλyt=n=1∞k=0n−12λn−2k−1−1ky2k+1S1,λn,2k+1tnn!,(51)cosλyt=n=0∞k=0n2λn−2k−1ky2kS1,λn,2ktnn!.We give the following theorem.Theorem 15.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(52)Bn,λk1,k2,⋯,kr;Sx,y=l=1nk=0l−12nlBn−l,λk1,k2,⋯,krxλl−2k−1−1ky2k+1S1,λl,2k+1,Bn,λk1,k2,⋯,kr;Cx,y=l=1nk=0l2nlBn−l,λk1,k2,⋯,krxλl−2k−1−1ky2k+1S1,λl,2k+1,where the notation · is Gauss’ notation and represents the maximum integer which does not exceed a number in the square bracket.Proof.
By (41)–(51), we observe that
(53)∑n=0∞Bn,λk1,k2,⋯,kr;Sx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtsinλyt=∑n=0∞Bn,λk1,k2,⋯,krxtnn!n=1∞k=0n−12λn−2k−1−1ky2k+1S1,λn,2k+1tnn!=n=0∞l=1nnlBn−l,λk1,k2,⋯,krxk=0l−12λl−2k−1−1ky2k+1S1,λl,2k+1tnn!,∑n=0∞Bn,λk1,k2,⋯,kr;Cx,ytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1reλxtcosλyt=∑n=0∞Bn,λk1,k2,⋯,krxtnn!n=1∞k=0n2λn−2k−1ky2kS1,λn,2ktnn!=n=0∞l=1nnlBn−l,λk1,k2,⋯,krxk=0l2λl−2k−1ky2kS1,λl,2ktnn!.
So, the proof is completed.We give the following proposition.Proposition 16.
The following relations(54)Bn,λk1,k2,⋯,kr;Sx+u,y=l=0nnlBn−l,λk1,k2,⋯,kr;Sx,yul,λ,Bn,λk1,k2,⋯,kr;Cx+u,y=l=0nnlBn−l,λk1,k2,⋯,kr;Cx,yul,λ.hold for k1,k2,⋯,kr∈ℤ,u∈ℂ and n≥0.Proof.
The proofs of this proposition can be done by utilizing the same proof methods used in Theorem7. So, we omit the proofs.Upon settingx=0 in (41) and (42), we consider the degenerate multi-poly-sine-Bernoulli polynomials Bn,λk1,k2,⋯,kr;Sy and the degenerate multi-poly-cosine-Bernoulli polynomials Bn,λk1,k2,⋯,kr;Cy as follows
(55)∑n=0∞Bn,λk1,k2,⋯,kr;Sytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1rsinλyt,(56)∑n=0∞Bn,λk1,k2,⋯,kr;Cytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1rcosλyt.We now provide the following theorem.Theorem 17.
Fork1,k2,⋯,kr∈ℤ and n>0, we have
(57)Bn,λk1,k2,⋯,kr;Cx,y=∑m=0n∑l=0mnmxlS2,λm,lBn−m,λk1,k2,⋯,kr;CywithBn,λk1,k2,⋯,kr;Cx,y=0.Proof.
The proofs of this theorem can be done by utilizing the same proof methods in Theorem9.Letα be any fixed real (or complex) number. The Bernoulli polynomials of order α is defined by (cf. [25])
(58)tet−1αext=∑n=0∞Bnαxtnn!,t<2π.When x=0, the Bernoulli polynomials of order α reduce to the Bernoulli numbers of order α, denoted by Bnα.We give the following relation.Theorem 18.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(59)Bn,λk1,k2,⋯,kr;S1,y−Bn,λk1,k2,⋯,kr;Sy=n∑l=0n−1n−1lBn−1−l,λk1,k2,⋯,kr;SyBl−1,(60)Bn,λk1,k2,⋯,kr;C1,y−Bn,λk1,k2,⋯,kr;Cy=n∑l=0n−1n−1lBn−1−l,λk1,k2,⋯,kr;CyBl−1.Proof.
By (55) and (56), we acquire
(61)∑n=0∞Bn,λk1,k2,⋯,kr;S1,ytnn!−∑n=0∞Bn,λk1,k2,⋯,kr;Sytnn!=r!Eik1,k2,⋯,kr;λlogλ1+teλt−1rsinλyteλt−1=∑n=0∞Bn,λk1,k2,⋯,kr;Sytn+1n!∑n=0∞Bn−1tnn!=∑n=0∞∑l=0nnlBn−l,λk1,k2,⋯,kr;SyBl−1tn+1n!.Thus, (59) is proved. We prove (60) in the same way.Here is a special case of Theorem18.Corollary 19.
Fork1,k2,⋯,kr∈ℤ and n≥0, we have
(62)Bn,λk1,k2,⋯,kr1−Bn,λk1,k2,⋯,kr=n∑l=0n−1n−1lBn−1−l,λk1,k2,⋯,krBl−1,which is a relation including the degenerate multi-poly-Bernoulli polynomials.


## 4. Conclusions

In this paper, we defined the degenerate multi-poly-Bernoulli polynomials by employing the degenerate multiple 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 multi-poly-Bernoulli polynomials. In the last section, we have defined degenerate multi-poly-Bernoulli polynomials of complex variables and then we have derived several properties and relations.



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