### Research Article

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

### G. Muhiuddin ,

^{1}### W. A. Khan,

^{2}### U. Duran ,

^{3}### and D. Al-Kadi

^{4}1Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

2Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia

3Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey

4Department of Mathematics and Statistic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Correspondence should be addressed to G. Muhiuddin; chishtygm@gmail.com

Received 17 April 2021; Accepted 18 May 2021; Published 2 June 2021 Academic Editor: Gangadharan Murugusundaramoorthy

Copyright © 2021 G. Muhiuddin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

### 1. Introduction

For any *λ ∈ ℝ/f0g (or ℂ/f0g), degenerate version of the*
*exponential function e*^{x}_{λ}*ðtÞ is deﬁned as follows (see [1–15])*

*e*^{x}* _{λ}*ð Þ

*t*

*≔ 1 + λt*ð Þ

^{x}*= 〠*

^{λ}^{∞}

*n=0*

ð Þ*x* _{n,}_{λ}*t*^{n}

*n*!, ð1Þ

*where ðxÞ*_{0,λ}*= 1 and ðxÞ*_{n,λ}*= xðx− λÞ ⋯ ðx − ðn − 1ÞλÞ for n*

≥ 1, (cf. [1–15]). It follows from (1) is lim_{λ→0}*e*^{x}_{λ}*ðtÞ = e*^{xt}. Note
*that e*^{1}_{λ}*ðtÞ ≔ e*_{λ}*ðtÞ:*

Carlitz [1] introduced the degenerate Bernoulli polyno- mials as follows:

*t*

*e** _{λ}*ð Þ

*t*− 1

*e*

^{x}*ð Þ =*

_{λ}*t*〠

^{∞}

*n=0*

*β**n*ð*x ;λ*Þ*t*^{n}

*n!:* ð2Þ

*Upon setting x = 0,* *β**n**ð0 ; λÞ ≔ β**n**ðλÞ are called the*
degenerate Bernoulli numbers.

Note that

lim*λ→0**β**n*ð*x ;λÞ = B**n*ð Þ,*x* ð3Þ

*where B*_{n}*ðxÞ are the familiar Bernoulli polynomials (cf. [1, 3,*
4, 6, 8, 11, 12, 14, 16–22])

*t*

*e** ^{t}*− 1

*e*

*= 〠*

^{xt}^{∞}

*n=0*

*B** _{n}*ð Þ

*x*

*t*

^{n}*n!*, ðj j < 2π*t* Þ: ð4Þ

*For k∈ ℤ, the polyexponential function Ei**k**ðxÞ is deﬁned*
by (see [21])

Ei* _{k}*ð Þ =

*x*〠

^{∞}

*n=1*

*x*^{n}*n*− 1

ð Þ!n^{k}*, k*ð *∈ ℤ*Þ: ð5Þ

*Setting k = 1 in (5), we have Ei*_{1}*ðxÞ = e*^{x}*− 1:*

Volume 2021, Article ID 7172054, 8 pages https://doi.org/10.1155/2021/7172054

The degenerate modiﬁed polyexponential function [12]

is deﬁned, for k ∈ ℤ and jxj < 1, by

Ei*k;**λ*ð Þ =*x* 〠^{∞}

*n=1*

ð Þ1 _{n,}_{λ}*n*− 1

ð Þ!n^{k}*x*^{n}*:* ð6Þ

Note that Ei_{1;λ}*ðxÞ = e*_{λ}*ðxÞ − 1:*

*Let k∈ ℤ and λ ∈ ℝ. The degenerate poly-Bernoulli poly-*
nomials, cf. [12], are deﬁned by

Ei_{k;}* _{λ}*ðlog

*ð*

_{λ}*1 + t*ÞÞ

*e** _{λ}*ð Þ

*t*− 1

*e*

^{x}*ð Þ =*

_{λ}*t*〠

^{∞}

*n=0*

*B*^{ð Þ}_{n,}^{k}* _{λ}*ð Þ

*x*

*t*

^{n}*n!*, ð7Þ

log* _{λ}*ð

*1 + t*Þ = 〠

^{∞}

*n=1*

*λ*^{n}^{−1}ð Þ1 _{n,}^{1}

*λ*

*t*^{n}

*n!*,ð*λ ∈ ℝ*Þ, ð8Þ
where log_{λ}*ð1 + tÞ are called the degenerate version of the log-*
arithm function (cf. [8, 12]), which is also the inverse func-
*tion of the degenerate exponential function e*_{λ}*ðtÞ as shown*
below (cf. [8])

*e** _{λ}*ðlog

*ð*

_{λ}*1 + t*ÞÞ = log

*ð*

_{λ}*e*

*ð*

_{λ}*1 + t*Þ

*Þ = 1 + t:*ð9Þ

*Letting x = 0 in (7), B*

^{ðkÞ}

_{n,λ}*ð0Þ ≔ B*

^{ðkÞ}*are called the type 2 degen- erate poly-Bernoulli numbers.*

_{n,λ}The degenerate Stirling numbers of the ﬁrst kind (cf.

[8, 13]) and second kind (cf. [4–6, 9, 17]) are deﬁned, respectively, by

1

*k!*ðlog* _{λ}*ð

*1 + t*ÞÞ

^{k}= 〠

^{∞}

*n=k*

*S*_{1,}* _{λ}*ð

*n, k*Þ

*t*

^{n}*n!, k*ð ≥ 0Þ, ð10Þ
and (cf. [1–27])

1

*k*!ð*e** _{λ}*ð Þ

*t*− 1Þ

^{k}= 〠

^{∞}

*n=k*

*S*_{2,λ}ð*n, k*Þ*t*^{n}

*n*!*, k*ð ≥ 0Þ: ð11Þ
Note that lim* _{λ→0}* in (10) and (1.8), we have (cf. [8, 13])

*log 1 + t*ð Þ

ð Þ^{k}

*k!* = 〠^{∞}

*n=k*

*S*1ð*n, k*Þ*t*^{n}

*n!*ð*k*≥ 0Þ, ð12Þ

and (cf. [4–6, 9, 17, 24])
*e** ^{t}*− 1
ð Þ

^{k}*k*! = 〠^{∞}

*n=k*

*S*_{2}ð*n, k*Þ*t*^{n}

*n*!ð*k*≥ 0Þ, ð13Þ
*where S*_{1}*ðn, kÞ and S*_{2}*ðn, kÞ are called the Stirling numbers*
of the ﬁrst kind and second kind.

The following paper is as follows. In Section 2, we deﬁne the degenerate multi-poly-Bernoulli polynomials and num- bers by using the degenerate multiple polyexponential func- tions 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

*Let k*_{1}*, k*_{2},*⋯, k**r**∈ ℤ. The degenerate multiple polyexponen-*
tial function Ei_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}_{;λ}*ðxÞ is deﬁned (cf. [15]) by*

Ei_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}_{;λ}ð Þ =*x* 〠

*0<n*_{1}*<n*_{2}<*⋯<n**r*

ð Þ1 _{n}

1,*λ*⋯ 1ð Þ_{n}

*r*,*λ**x*^{n}^{r}*n*_{1}− 1

ð Þ!⋯ nð *r*− 1Þ!n* ^{k}*1

^{1}

*⋯ n*

^{k}*r*

^{r}, ð14Þ

*where the sum is over all integers n*_{1}*, n*_{2},*⋯, n**r* satisfying
*0 < n*_{1}*< n*_{2}<*⋯ < n**r*. Utilizing this function, Kim et al. [15]

introduced and studied the degenerate multi-poly-Genocchi polynomials given by

2* ^{r}*Ei

_{k}_{1}

_{,k}_{2}

_{,⋯,k}

_{r}_{;λ}ðlog

*ð*

_{λ}*1 + t*ÞÞ

*e*

*ð Þ + 1*

_{λ}*t*

ð Þ^{r}*e*^{x}* _{λ}*ð Þ =

*t*〠

^{∞}

*n=0*

*g*^{ð}_{n,}^{k}_{λ}^{1}^{,k}^{2}^{,}^{⋯k}^{r}^{Þ}ð Þ*x* *t*^{n}*n!: ð15Þ*

Inspired by the deﬁnition of degenerate multi-poly-Genocchi polynomials, using the degenerate multiple polyexponential function (14), we give the following deﬁnition.

Deﬁnition 1. Let k1*, k*_{2},*⋯, k**r**∈ ℤ and λ ∈ ℝ, we consider the*
degenerate multi-poly-Bernoulli polynomials are given by

*r*!Ei*k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þ =

*t*〠

^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}*n*!*:*

ð16Þ

*Upon setting x = 0 in (16), the degenerate multi-poly-*
Bernoulli polynomials reduce to the corresponding numbers,
namely, the type 2 degenerate multi-poly-Bernoulli numbers
B^{ðk}*n,**λ*^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð0Þ ≔ B^{ðk}*n,**λ*^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}.

Remark 2. As *λ → 0, the degenerate multi-poly-Bernoulli*
polynomials reduce to the multi-poly-Bernoulli polynomials
given by

*r*!Ei*k*_{1}*,k*_{2},*⋯,k**r*ð*log 1 + t*ð ÞÞ
*e** ^{t}*− 1

ð Þ^{r}*e** ^{xt}*= 〠

^{∞}

*n=0*

B^{ð}*n*^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}

*n*!*: ð17Þ*

*Remark 3. Upon setting r = 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 degen- erate multi-poly-Bernoulli polynomials, we ﬁrst give the following result.

**Proposition 4 (Derivative Property). For k***1**, k** _{2}*,

*⋯, k*

*r*

*∈ ℤ*and

*λ ∈ ℝ, we have*

*d*

*dxEi*_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}_{;λ}ð Þ =*x* *1*

*xEi*_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}* _{−1;λ}*ð Þ:

*x*ð18Þ Proof. By (14), we see that

*d*

*dx*Ei_{k}_{1}_{,k}_{2}_{,}_{⋯,k}_{r}_{;}* _{λ}*ð Þ =

*x*

*d*

*dx* 〠

*0<n*1*<n*2<⋯<n*r*

ð Þ1 _{n}

1,*λ*⋯ 1ð Þ_{n}

*r*,*λ**x*^{n}^{r}*n*_{1}− 1

ð Þ*!⋯ n*ð *r*− 1Þ*!n** ^{k}*1

^{1}

*⋯ n*

^{k}

^{r}

^{r}=1

*x* 〠

*0<n*_{1}*<n*_{2}<⋯<n*r*

ð Þ1 _{n}_{1}_{,λ}⋯ 1ð Þ_{n}_{r}_{,λ}*x*^{n}^{r}*n*1− 1

ð Þ!⋯ nð *r*− 1Þ!n* ^{k}*1

^{1}

*⋯ n*

^{k}*r*

^{r}^{−1}

=1

*x*Ei_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}* _{−1;λ}*ð Þ:

*x*

ð19Þ

**Theorem 5. The following relationship**

B^{ð}*n,*^{k}*λ*^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð Þ =*x* 〠^{n}

*j=0*

*n*
*j*
!

B^{ð}*n*^{k}*−j,λ*^{1}^{,k}^{2}^{,}^{⋯k}^{r}^{Þ}ð Þ*x* _{j,}* _{λ}*, ð20Þ

*holds for n≥ 0.*

Proof. Recall Deﬁnition 1 that

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}

*n*!=*r!Ei**k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þ

*t*

= 〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}*t*^{n}*n*!〠^{∞}

*j=0*

ð Þ*x* _{j,}_{λ}*t*^{m}*m*!

= 〠^{∞}

*n=0*

〠^{n}

*j=0*

*n*
*j*
!

B^{ð}_{n−j,λ}^{k}^{1}^{,k}^{2}^{,⋯k}^{r}^{Þ}ð Þ*x* _{j,}_{λ}

!*t*^{n}*n*!,

ð21Þ

which gives the asserted result (20).

*The degenerate Bernoulli polynomials of order r are*
given by the following series expansion:

〠^{∞}

*n=0*

*β*^{ð Þ}_{n}* ^{r}*ð

*x ;λ*Þ

*t*

^{n}*n*!= *t*

*e** _{λ}*ð Þ

*t*− 1

*r*

*e*^{x}* _{λ}*ð Þ,

*t*ð22Þ

(cf. [3, 6, 8, 17]).

We provide the following theorem.

**Theorem 6. For n ≥ r. Then**

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ =*x* * ^{n+r}*〠

*m=0*

〠

*0<n*_{1}*<n** _{2}*<

*⋯<n*

*r*

*≤m*

*n + r*
*m*

!

*β*^{ð Þ}*n+r*^{r}*−m*ð*x ;λÞS**1**:λ*ð*m, n** _{r}*Þ

× *n!r!n**r**! 1*ð Þ_{n}_{1}_{,λ}*⋯ 1*ð Þ_{n}_{r}_{,λ}
*n + r*

ð Þ! nð *1**− 1*Þ!⋯ nð *r**− 1*Þ!n^{k}*1*^{1}*⋯ n*^{k}*r*^{r}

*:*
ð23Þ

Proof. Recall from Deﬁnition 1 and (10) that

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}

*n!*= *r!e*^{x}* _{λ}*ð Þ

*t*

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* 〠

*0<n*1*<n*2<⋯<n_{r}

ð Þ1_{n}

1,λ⋯ 1ð Þ_{n}

*r*,λðlog* _{λ}*ð

*1 + t*ÞÞ

^{n}

^{r}*n*

_{1}− 1

ð Þ!⋯ nð * _{r}*− 1Þ!n

^{k}_{1}

^{1}

*⋯ n*

^{k}*r*

^{r}= *r!e*^{x}* _{λ}*ð Þ

*t*

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* 〠

*0<n*1*<n*2<⋯<n_{r}

ð Þ1_{n}

1,λ⋯ 1ð Þ_{n}

*r*,λ*n** _{r}*!

*n*

_{1}− 1

ð Þ!⋯ nð * _{r}*− 1Þ!n

^{k}_{1}

^{1}

*⋯ n*

^{k}*r*

^{r} 〠^{∞}

*m=n**r*

*S*_{1:λ}ð*m, n** _{r}*Þ

*t*

^{m}*m!*

=*r!*

*t*^{r}

*t*^{r}*e*^{x}* _{λ}*ð Þ

*t*

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}

〠^{∞}

*m=n**r*

〠

*0<n*1*<n*2<⋯<n*r**≤m*

ð Þ1_{n}

1,λ⋯ 1ð Þ_{n}

*r*,λ*S*_{1:λ}ð*m, n*_{r}*Þn** _{r}*!

*n*

_{1}− 1

ð Þ!⋯ nð * _{r}*− 1Þ!n

^{k}_{1}

^{1}

*⋯ n*

^{k}*r*

^{r}!

*t*^{m}*m!*=*r!*

*t** ^{r}*〠

^{∞}

*l=0*

*β*^{ð Þ}_{l}* ^{r}*ð

*x ;*

*λ*Þ

*t*

^{l}*l!*〠

^{∞}

*m=n**r*

〠

*0<n*1*<n*2<⋯<n*r**≤m*

ð Þ1_{n}

1,λ⋯ 1ð Þ_{n}

*r*,λ*S*_{1:λ}ð*m, n*_{r}*Þn** _{r}*!

*n*

_{1}− 1

ð Þ!⋯ nð * _{r}*− 1Þ!n

^{k}_{1}

^{1}

*⋯ n*

^{k}*r*

^{r}!

*t*^{m}*m!*= 〠^{∞}

*n=r*

〠^{n}

*m=0*

*n*
*m*
!

〠

*0<n*1*<n*2<⋯<n*r**≤m*

*r!n** _{r}*! 1ð Þ

_{n}1,λ⋯ 1ð Þ_{n}

*r*,λ

*n*_{1}− 1

ð Þ!⋯ nð * _{r}*− 1Þ!n

^{k}_{1}

^{1}

*⋯ n*

^{k}*r*

^{r}* β*^{ð Þ}_{n−m}* ^{r}* ð

*x ;*

*λ*

*ÞS*

_{1:λ}ð

*m, n*

*Þ*

_{r}*t*

^{n−r}*n!*,

ð24Þ

which means the claimed result (23).

**Theorem 7. The following formula**

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x + y*Þ =〠^{n}

*j=0*

*n*
*j*
!

ð Þ*y* _{j,}* _{λ}*B

^{ð}

_{n−j,λ}

^{k}

^{1}

^{,k}

^{2}^{,⋯,k}

^{r}^{Þ}ð Þ,

*x*ð25Þ

*is valid for k*_{1}*, k** _{2}*,

*⋯, k*

*r*

*∈ ℤ and n ≥ 0.*

Proof. In view of Deﬁnition 1, we see that

〠^{∞}

*n=0*

B^{ð}*n,*^{k}*λ*^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð*x + y*Þ*t*^{n}

*n!*=*r!Ei**k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x+y}* _{λ}* ð Þ

*t*

=〠^{∞}

*i=0*

B^{ð}_{i,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{i}*i*!〠^{∞}

*j=0*

ð Þ*y* _{j,}_{λ}*t*^{m}*m*!

= 〠^{∞}

*n=0*

〠^{n}

*j=0*

*n*
*j*
!

ð Þ*y* * _{m,λ}*B

^{ð}

*n*

^{k}*−j,λ*

^{1}

^{,k}^{2}

^{,⋯,k}

^{r}^{Þ}ð Þ

*x*

!*t*^{n}*n!*,
ð26Þ

which implies the desired result (25).

**Theorem 8. The following relation**

*d*

*dx*B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ=*x* ^{n}_{l=1}*n*
*l*
!

B^{ð}_{n−l,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* ð Þ*−λ** ^{l−1}*ð

*l− 1*Þ!, ð27Þ

*is valid for k*_{1}*, k** _{2}*,

*⋯, k*

*r*

*∈ ℤ and n ≥ 0.*

Proof. To investigate the derivative property ofB^{ðk}*n,**λ*^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}*ðxÞ*
that

〠^{∞}

*n=0*

*d*

*dx*B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}

*n!*= *r!Ei*_{k}_{1}_{,k}_{2}_{,⋯,k}_{r}_{;λ}ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}

*d*
*dx**e*^{x}* _{λ}*ð Þ

*t*

= 〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}*n!*

1

*λ*ln 1 +ð *λt*Þ

= 〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}*n!*

!

〠^{∞}

*l=1*

−1
ð Þ^{l+1}

*l* *λ*^{l−1}*t*^{l}

= 〠^{∞}

*n=0*

〠^{∞}

*l=1*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð Þ*x* ð Þ−1^{l+1}*l* *λ*^{l−1}*t*^{n+l}

*n!* ,
ð28Þ

which provides the asserted result (27).

We here give a relation including the degenerate multi- poly-Bernoulli polynomials with numbers and the degener- ate Stirling numbers of the second kind.

**Theorem 9. The following correlation**

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ =*x* 〠^{n}

*m=0*

〠^{m}

*l=0*

*n*
*m*
!

ð Þ*x* _{l}*S** _{2,λ}*ð

*m, l*ÞB

^{ð}

_{n−m,λ}

^{k}

^{1}

^{,k}

^{2}^{,⋯,k}

^{r}^{Þ}, ð29Þ

*is valid for k*_{1}*, k** _{2}*,

*⋯, k*

*r*

*∈ ℤ and n ≥ 0.*

Proof. By means of Deﬁnition 1, we attain that

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð Þ*x* *t*^{n}

*n!*=*r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þ

*t*

=*r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* ð

*e*

*ð Þ*

_{λ}*t*− 1 + 1Þ

^{x}=*r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* 〠

^{∞}

*l=0*

*x*
*l*
!

*e** _{λ}*ð Þ

*t*− 1

ð Þ^{l}

=〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}*t*^{n}*n!*〠^{∞}

*l=0*

ð Þ*x** _{l}*〠

^{∞}

*m=l*

*S*_{2,λ}ð*m, l*Þ*t*^{m}*m!*

=〠^{∞}

*n=0*

〠^{n}

*m=0*

〠^{m}

*l=0*

*n*
*m*
!

ð Þ*x* _{l}*S*_{2,λ}ð*m, l*ÞB^{ð}_{n−m,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}

!*t*^{n}*n!*,

ð30Þ

*where the notation ðxÞ** _{l}* is falling factorial that is deﬁned

*by ðxÞ*

_{0}

*= 1 and ðxÞ*

_{n}*= xðx− 1Þ ⋯ ðx − ðn − 1ÞÞ 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

*e*^{m}* _{λ}*ð Þ

*t*− 1

ð Þ^{k}

*m*^{k}*k*! *e*^{α}* _{λ}*ð Þ =

*t*〠

^{∞}

*n=k*

*W*_{m,}* _{α}*ð

*n, k*j

*λ*Þ

*t*

^{n}*n!, k*ð ≥ 0Þ: ð31Þ

Kim also provided several correlations including the degen- erate Stirling numbers of the second kind and the degenerate Whitney numbers (see [5]).

We now give a correlation as follows.

**Theorem 10. For k***1**, k** _{2}*,

*⋯k*

*r*

*∈ ℤ and n ≥ 0, we have*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*xu +**α*Þ = 〠^{n}

*m=0*

〠^{m}

*l=0*

*n*
*m*
!

*u** ^{l}*ð Þ

*x*

_{l}*W*

_{u,}*ð*

_{α}*m, l*j

*λ*ÞB

^{ð}

_{n−m,λ}

^{k}

^{1}

^{,k}

^{2}^{,⋯,k}

^{r}^{Þ}

*:*ð32Þ

Proof. Using (31) and Deﬁnition 1, we acquire that

〠^{∞}

*n=0*

B^{ð}*n,*^{k}*λ*^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð*xu +α*Þ*t*^{n}*n!*

=*r!Ei**k*_{1}*,k*_{2},⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{α}_{λ}*ð Þet* ^{xu}* _{λ}* ð Þ

*t*

= *r!Ei**k*_{1}*,k*_{2},⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{α}_{λ}*ð Þ et* ð^{u}* _{λ}*ð Þ

*t*− 1 + 1Þ

^{x}=*r!Ei**k*_{1}*,k*_{2},⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{α}* _{λ}*ð Þ

*t*〠

^{∞}

*l=0*

*x*
*l*
!

*e*^{u}* _{λ}*ð Þ

*t*− 1

ð Þ^{l}

=*r*!Ei*k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* 〠

^{∞}

*l=0*

*u** ^{l}*ð Þ

*x*

*ð*

_{l}*e*

^{u}*ð Þ*

_{λ}*t*− 1Þ

^{l}*l!u*

^{l}*e*

^{α}*ð Þ*

_{λ}*t*

= *r*!Ei*k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ* ^{r}* 〠

^{∞}

*l=0*

*u** ^{l}*ð Þ

*x*

*ð*

_{l}*e*

^{u}*ð Þ*

_{λ}*t*− 1Þ

^{l}*l!u*

^{l}*e*

^{α}*ð Þ*

_{λ}*t*

= 〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}*t*^{n}*n!*〠^{∞}

*n=0*

〠^{n}

*l=0*

*u** ^{l}*ð Þ

*x*

_{l}*W*

*ð*

_{u,α}*n, l*j

*λ*Þ

*t*

^{n}*n!*

= 〠^{∞}

*n=0*

〠^{n}

*m=0*

〠^{m}

*l=0*

*n*
*m*
!

*u** ^{l}*ð Þ

*x*

_{l}*W*

*ð*

_{u,α}*m, l*j

*λ*ÞB

^{ð}

_{n−m,λ}

^{k}^{1}

^{,k}^{2}

^{,⋯,k}

^{r}^{Þ}

!*t*^{n}*n!*,

ð33Þ which implies the asserted result (32).

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

In [25], Kim et al. deﬁned the degenerate sine sin_{λ}*t and*
cosine cos_{λ}*t functions by*

sin^{ð Þ}_{λ}* ^{x}*ð Þ =

*t*

*e*

^{ix}*ð Þ*

_{λ}*t*

*− e*

^{−ix}*ð Þ*

_{λ}*t*

*2i* and cos^{ð Þ}_{λ}* ^{x}*ð Þ =

*t*

*e*

^{ix}

_{λ}*ð Þ + et*

^{−ix}*ð Þ*

_{λ}*t*

2 ,

ð34Þ

*where i =* ﬃﬃﬃﬃﬃﬃ
p−1

. Note that lim* _{λ→0}*sin

^{ðxÞ}

_{λ}*ðtÞ = sin xt and*lim

*cos*

_{λ→0}

^{ðxÞ}

_{λ}*ðtÞ = cos xt. From (34), it is readily that*

*e*^{ix}* _{λ}*ð Þ = cos

*t*

^{ð Þ}

_{λ}

^{x}*ð Þ + isint*

^{ð Þ}

_{λ}*ð Þ*

^{x}*t*

*:*ð35Þ By these functions in (34), the degenerate sine-

*polynomials S*

_{k,λ}*ðx, yÞ and degenerate cosine-polynomials*

*C*

_{k,λ}*ðx, yÞ are introduced (cf. [25]) by*

〠^{∞}

*n=0*

*S*_{k,}* _{λ}*ð

*x, y*Þ

*t*

^{n}*n!* *= e*^{x}* _{λ}*ð Þsin

*t*

^{ð Þ}

_{λ}*ð Þ,*

^{y}*t*ð36Þ

〠^{∞}

*n=0*

*C*_{k,}* _{λ}*ð

*x, y*Þ

*t*

^{n}*n!= e*^{x}* _{λ}*ð Þcos

*t*

^{ð Þ}

_{λ}*ð Þ:*

^{y}*t*ð37Þ 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 investi- gate 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. Let k1*, k*_{2},*⋯, k**r**∈ ℤ. We deﬁne a new form of*
the degenerate multi-poly-Bernoulli polynomials of complex
variable by the following generating function:

*r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ e

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x+iy}* _{λ}* ð Þ =

*t*〠

^{∞}

*n=0*

B^{ð}_{n,}^{k}_{λ}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{Þ}ð*x + iy*Þ*t*^{n}*n!:*
ð38Þ

By (34) and (38), we observe that

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x + iy*Þ− B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x− iy*Þ

*2i*

*t*^{n}*n*!

= *r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þsin

*t*

^{ð Þ}

_{λ}*ð Þ,*

^{y}*t*

ð39Þ

and

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x + iy*Þ + B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x− iy*Þ

2

*t*^{n}*n!*

= *r!Ei**k*1*,k*2,⋯,k*r*;λðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þcos

*t*

^{ð Þ}

_{λ}*ð Þ*

^{y}*t*

*:*

ð40Þ

In view of (39) and (40), we consider the degenerate
multi-poly-sine-Bernoulli polynomials B^{ðk}_{n,λ}^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{;SÞ}*ðx, yÞ*
with two parameters and the degenerate multi-poly-cosine-
Bernoulli polynomials B^{ðk}_{n,}_{λ}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;CÞ}*ðx, yÞ with two parame-*
ters as follows:

〠^{∞}

*n=0*

B^{ð}_{n,}^{k}_{λ}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;S}^{Þ}ð*x, y*Þ*t*^{n}

*n!*=*r!Ei**k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þsin

*t*

^{ð Þ}

_{λ}*ð Þ,*

^{y}*t*ð41Þ

〠^{∞}

*n=0*

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;C}^{Þ}ð*x, y*Þ*t*^{n}

*n*!=*r*!Ei*k*_{1}*,k*_{2},*⋯,k**r*;*λ*ðlog* _{λ}*ð

*1 + t*ÞÞ

*e*

*ð Þ*

_{λ}*t*− 1

ð Þ^{r}*e*^{x}* _{λ}*ð Þcos

*t*

^{ð Þ}

_{λ}*ð Þ:*

^{y}*t*ð42Þ Note that

*λ→0*limB^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;S}^{Þ}ð*x, y*Þ≔ B^{ð}*n*^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;S}^{Þ}ð*x, y*Þ and

lim*λ→0*B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;C}^{Þ}ð*x, y*Þ≔ B^{ð}*n*^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;C}^{Þ}ð*x, y*Þ, ð43Þ

which are multi-poly-sine-Bernoulli polynomials
B^{ðk}*n*^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{;SÞ}*ðx, yÞ and multi-poly-cosine-Bernoulli polyno-*
mials B^{ðk}*n*^{1}^{,k}^{2}^{,}^{⋯,k}^{r}^{;CÞ}*ðx, yÞ with two parameters.*

By (39)-(42), we see that

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;S}^{Þ}ð*x, y*Þ =B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x + iy*Þ− B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x− iy*Þ

*2i* ,

B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{;C}^{Þ}ð*x, y*Þ =B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x + iy*Þ + B^{ð}_{n,λ}^{k}^{1}^{,k}^{2}^{,⋯,k}^{r}^{Þ}ð*x− iy*Þ

2 *:*

ð44Þ We now give the two summation formulae by the follow- ing theorem.