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

## Full text

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1

2

### U. Duran ,

3

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

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 exλðtÞ is deﬁned as follows (see [1–15])

exλð Þt ≔ 1 + λtð Þxλ= 〠

n=0

ð Þx n,λtn

n!, ð1Þ

where ðxÞ0,λ= 1 and ðxÞn,λ= xðx− λÞ ⋯ ðx − ðn − 1ÞλÞ for n

≥ 1, (cf. [1–15]). It follows from (1) is limλ→0exλðtÞ = ext. Note that e1λðtÞ ≔ eλðtÞ:

Carlitz  introduced the degenerate Bernoulli polyno- mials as follows:

t

eλð Þt − 1exλð Þ =t

n=0

βnðx ;λÞtn

n!: ð2Þ

Upon setting x = 0, βnð0 ; λÞ ≔ βnðλÞ are called the degenerate Bernoulli numbers.

Note that

limλ→0βnðx ;λÞ = Bnð Þ,x ð3Þ

where BnðxÞ are the familiar Bernoulli polynomials (cf. [1, 3, 4, 6, 8, 11, 12, 14, 16–22])

t

et− 1ext= 〠

n=0

Bnð Þx tn

n!, ðj j < 2πt Þ: ð4Þ

For k∈ ℤ, the polyexponential function EikðxÞ is deﬁned by (see )

Eikð Þ =x

n=1

xn n− 1

ð Þ!nk, kð ∈ ℤÞ: ð5Þ

Setting k = 1 in (5), we have Ei1ðxÞ = ex− 1:

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

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The degenerate modiﬁed polyexponential function 

is deﬁned, for k ∈ ℤ and jxj < 1, by

Eik;λð Þ =x

n=1

ð Þ1 n,λ n− 1

ð Þ!nkxn: ð6Þ

Note that Ei1;λðxÞ = eλðxÞ − 1:

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

Eik;λðlogλð1 + tÞÞ

eλð Þt − 1 exλð Þ =t

n=0

Bð Þn,kλð Þx tn

n!, ð7Þ

logλð1 + tÞ = 〠

n=1

λn−1ð Þ1 n,1

λ

tn

n!λ ∈ ℝÞ, ð8Þ where logλð1 + 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 eλðtÞ as shown below (cf. )

eλðlogλð1 + tÞÞ = logλðeλð1 + tÞÞ = 1 + t: ð9Þ Letting x = 0 in (7), BðkÞn,λð0Þ ≔ BðkÞn,λare called the type 2 degen- erate 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

1

k!ðlogλð1 + tÞÞk= 〠

n=k

S1,λðn, kÞtn

n!, kð ≥ 0Þ, ð10Þ and (cf. [1–27])

1

keλð Þt − 1Þk= 〠

n=k

S2,λðn, kÞtn

n!, kð ≥ 0Þ: ð11Þ Note that limλ→0 in (10) and (1.8), we have (cf. [8, 13])

log 1 + tð Þ

ð Þk

k! = 〠

n=k

S1ðn, kÞtn

n!ðk≥ 0Þ, ð12Þ

and (cf. [4–6, 9, 17, 24]) et− 1 ð Þk

k! = 〠

n=k

S2ðn, kÞtn

nk≥ 0Þ, ð13Þ where S1ðn, kÞ and S2ðn, 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 k1, k2,⋯, kr∈ ℤ. The degenerate multiple polyexponen- tial function Eik1,k2,⋯,krðxÞ is deﬁned (cf. ) by

Eik1,k2,⋯,krð Þ =x

0<n1<n2<⋯<nr

ð Þ1 n

1,λ⋯ 1ð Þn

r,λxnr n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

, ð14Þ

where the sum is over all integers n1, n2,⋯, nr satisfying 0 < n1< n2<⋯ < nr. Utilizing this function, Kim et al. 

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

2rEik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þ + 1t

ð Þr exλð Þ =t

n=0

gðn,kλ1,k2,⋯krÞð Þx tn 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, k2,⋯, kr∈ ℤ and λ ∈ ℝ, we consider the degenerate multi-poly-Bernoulli polynomials are given by

r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þ =t

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn 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 Bðkn,λ1,k2,⋯,krÞð0Þ ≔ Bðkn,λ1,k2,⋯,krÞ.

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

r!Eik1,k2,⋯,krðlog 1 + tð ÞÞ et− 1

ð Þr ext= 〠

n=0

Bðnk1,k2,⋯,krÞð Þx tn

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.

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Proposition 4 (Derivative Property). For k1, k2,⋯, kr∈ ℤ andλ ∈ ℝ, we have

d

dxEik1,k2,⋯,krð Þ =x 1

xEik1,k2,⋯,kr−1;λð Þ:x ð18Þ Proof. By (14), we see that

d

dxEik1,k2,⋯,kr;λð Þ =x d

dx

0<n1<n2<⋯<nr

 ð Þ1 n

1,λ⋯ 1ð Þn

r,λxnr n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

=1

x

0<n1<n2<⋯<nr

 ð Þ1 n1⋯ 1ð Þnrxnr n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr−1

=1

xEik1,k2,⋯,kr−1;λð Þ:x

ð19Þ

Theorem 5. The following relationship

Bðn,kλ1,k2,⋯,krÞð Þ =xn

j=0

n j !

Bðnk−j,λ1,k2,⋯krÞð Þx j,λ, ð20Þ

holds for n≥ 0.

Proof. Recall Deﬁnition 1 that

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn

n!=r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þt

= 〠

n=0

Bðn,λk1,k2,⋯,krÞtn n!〠

j=0

ð Þx j,λtm m!

= 〠

n=0

n

j=0

n j !

Bðn−j,λk1,k2,⋯krÞð Þx j,λ

!tn n!,

ð21Þ

which gives the asserted result (20).

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

n=0

βð Þnrðx ;λÞtn

n!= t

eλð Þt − 1

 r

exλð Þ,t ð22Þ

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

We provide the following theorem.

Theorem 6. For n ≥ r. Then

Bðn,λk1,k2,⋯,krÞð Þ =x n+r

m=0

0<n1<n2<⋯<nr≤m

 n + r m

!

βð Þn+rr −mðx ;λÞS1ðm, nrÞ

× n!r!nr! 1ð Þn1⋯ 1ð Þnr n + r

ð Þ! nð 1− 1Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

: ð23Þ

Proof. Recall from Deﬁnition 1 and (10) that

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn

n!= r!exλð Þt eλð Þt − 1

ð Þr

0<n1<n2<⋯<nr

ð Þ1n

1⋯ 1ð Þn

rðlogλð1 + tÞÞnr n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

= r!exλð Þt eλð Þt − 1

ð Þr

0<n1<n2<⋯<nr

 ð Þ1n

1⋯ 1ð Þn

rnr! n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

 〠

m=nr

S1:λðm, nrÞtm m!

=r!

tr

trexλð Þt eλð Þt − 1

ð Þr

 

m=nr



0<n1<n2<⋯<nr≤m

ð Þ1n

1⋯ 1ð Þn

rS1:λðm, nrÞnr! n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

!

tm m!=r!

tr

l=0

βð Þlrðx ;λÞtl l!

m=nr



0<n1<n2<⋯<nr≤m

ð Þ1n

1⋯ 1ð Þn

rS1:λðm, nrÞnr! n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

!

tm m!=

n=r

n

m=0

n m !

0<n1<n2<⋯<nr≤m

 r!nr! 1ð Þn

1⋯ 1ð Þn

r

n1− 1

ð Þ!⋯ nð r− 1Þ!nk11⋯ nkrr

 βð Þn−mr ðx ;λÞS1:λðm, nrÞtn−r n! ,

ð24Þ

which means the claimed result (23).

Theorem 7. The following formula

Bðn,λk1,k2,⋯,krÞðx + yÞ =〠n

j=0

n j !

ð Þy j,λBðn−j,λk1,k2,⋯,krÞð Þ,x ð25Þ

is valid for k1, k2,⋯, kr∈ ℤ and n ≥ 0.

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Proof. In view of Deﬁnition 1, we see that

n=0

Bðn,kλ1,k2,⋯,krÞðx + yÞtn

n!=r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr ex+yλ ð Þt

=

i=0

Bði,λk1,k2,⋯,krÞð Þx ti i!

j=0

ð Þy j,λtm m!

=

n=0

n

j=0

n j !

ð Þy m,λBðnk−j,λ1,k2,⋯,krÞð Þx

!tn n!, ð26Þ

which implies the desired result (25).

Theorem 8. The following relation

d

dxBðn,λk1,k2,⋯,krÞð Þ=x nl=1 n l !

Bðn−l,λk1,k2,⋯,krÞð Þx ð Þ−λl−1ðl− 1Þ!, ð27Þ

is valid for k1, k2,⋯, kr∈ ℤ and n ≥ 0.

Proof. To investigate the derivative property ofBðkn,λ1,k2,⋯,krÞðxÞ that

n=0

d

dxBðn,λk1,k2,⋯,krÞð Þx tn

n!= r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr

d dxexλð Þt

=

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn n!

1

λln 1 +ð λtÞ

=

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn n!

!

l=1

−1 ð Þl+1

l λl−1tl

=

n=0

l=1

Bðn,λk1,k2,⋯,krÞð Þx ð Þ−1l+1 l λl−1tn+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

Bðn,λk1,k2,⋯,krÞð Þ =xn

m=0

m

l=0

n m !

ð Þx lS2,λðm, lÞBðn−m,λk1,k2,⋯,krÞ, ð29Þ

is valid for k1, k2,⋯, kr∈ ℤ and n ≥ 0.

Proof. By means of Deﬁnition 1, we attain that

n=0

Bðn,λk1,k2,⋯,krÞð Þx tn

n!=r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þt

=r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr ðeλð Þt − 1 + 1Þx

=r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr

l=0

x l !

eλð Þt − 1

ð Þl

=

n=0

Bðn,λk1,k2,⋯,krÞtn n!

l=0

ð Þxl

m=l

S2,λðm, lÞtm m!

=

n=0

n

m=0

m

l=0

n m !

ð Þx lS2,λðm, lÞBðn−m,λk1,k2,⋯,krÞ

!tn n!,

ð30Þ

where the notation ðxÞl is falling factorial that is deﬁned by ðxÞ0= 1 and ðxÞn= xðx− 1Þ ⋯ ðx − ðn − 1ÞÞ for n ≥ 1, (cf. [1, 2, 5–14, 21, 23, 24]). So, the proof is completed.

Kim  introduced the degenerate Whitney numbers are given by

emλð Þt − 1

ð Þk

mkk! eαλð Þ =t

n=k

Wm,αðn, kjλÞtn

n!, kð ≥ 0Þ: ð31Þ

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

We now give a correlation as follows.

Theorem 10. For k1, k2,⋯kr∈ ℤ and n ≥ 0, we have

Bðn,λk1,k2,⋯,krÞðxu +αÞ = n

m=0

m

l=0

n m !

ulð Þx lWu,αðm, ljλÞBðn−m,λk1,k2,⋯,krÞ: ð32Þ

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

n=0

Bðn,kλ1,k2,⋯,krÞðxu +αÞtn n!

=r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr eαλð Þet xuλ ð Þt

= r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr eαλð Þ et ðuλð Þt − 1 + 1Þx

=r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr eαλð Þt

l=0

x l !

euλð Þt − 1

ð Þl

=r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr

l=0

ulð Þx lðeuλð Þt − 1Þl l!ul eαλð Þt

(5)

= r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr

l=0

ulð Þx lðeuλð Þt − 1Þl l!ul eαλð Þt

= 〠

n=0

Bðn,λk1,k2,⋯,krÞtn n!

n=0

n

l=0

ulð Þx lWu,αðn, ljλÞtn n!

= 〠

n=0

n

m=0

m

l=0

n m !

ulð Þx lWu,αðm, ljλÞBðn−m,λk1,k2,⋯,krÞ

!tn n!,

ð33Þ which implies the asserted result (32).

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

In , Kim et al. deﬁned the degenerate sine sinλt and cosine cosλt functions by

sinð Þλxð Þ =t eixλð Þt − e−ixλ ð Þt

2i and cosð Þλxð Þ =t eixλð Þ + et −ixλ ð Þt

2 ,

ð34Þ

where i = ﬃﬃﬃﬃﬃﬃ p−1

. Note that limλ→0sinðxÞλ ðtÞ = sin xt and limλ→0cosðxÞλ ðtÞ = cos xt. From (34), it is readily that

eixλð Þ = cost ð Þλxð Þ + isint ð Þλxð Þt : ð35Þ By these functions in (34), the degenerate sine- polynomials Sk,λðx, yÞ and degenerate cosine-polynomials Ck,λðx, yÞ are introduced (cf. ) by

n=0

Sk,λðx, yÞtn

n! = exλð Þsint ð Þλyð Þ,t ð36Þ

n=0

Ck,λðx, yÞtn

n!= exλð Þcost ð Þλyð Þ:t ð37Þ Several properties of these polynomials in (36) and (37) are studied and investigated in . Also, by means of these functions, Kim et al.  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, k2,⋯, kr∈ ℤ. We deﬁne a new form of the degenerate multi-poly-Bernoulli polynomials of complex variable by the following generating function:

r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr ex+iyλ ð Þ =t

n=0

Bðn,kλ1,k2,⋯,krÞðx + iyÞtn n!: ð38Þ

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

n=0

Bðn,λk1,k2,⋯,krÞðx + iyÞ− Bðn,λk1,k2,⋯,krÞðx− iyÞ

 

2i

tn n!

= r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þsint ð Þλyð Þ,t

ð39Þ

and

n=0

Bðn,λk1,k2,⋯,krÞðx + iyÞ + Bðn,λk1,k2,⋯,krÞðx− iyÞ

 

2

tn n!

= r!Eik1,k2,⋯,krðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þcost ð Þλyð Þt :

ð40Þ

In view of (39) and (40), we consider the degenerate multi-poly-sine-Bernoulli polynomials Bðkn,λ1,k2,⋯,kr;SÞðx, yÞ with two parameters and the degenerate multi-poly-cosine- Bernoulli polynomials Bðkn,λ1,k2,⋯,kr;CÞðx, yÞ with two parame- ters as follows:

n=0

Bðn,kλ1,k2,⋯,kr;SÞðx, yÞtn

n!=r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þsint ð Þλyð Þ,t ð41Þ

n=0

Bðn,λk1,k2,⋯,kr;CÞðx, yÞtn

n!=r!Eik1,k2,⋯,kr;λðlogλð1 + tÞÞ eλð Þt − 1

ð Þr exλð Þcost ð Þλyð Þ:t ð42Þ Note that

λ→0limBðn,λk1,k2,⋯,kr;SÞðx, yÞ≔ Bðnk1,k2,⋯,kr;SÞðx, yÞ and

limλ→0Bðn,λk1,k2,⋯,kr;CÞðx, yÞ≔ Bðnk1,k2,⋯,kr;CÞðx, yÞ, ð43Þ

which are multi-poly-sine-Bernoulli polynomials Bðkn1,k2,⋯,kr;SÞðx, yÞ and multi-poly-cosine-Bernoulli polyno- mials Bðkn1,k2,⋯,kr;CÞðx, yÞ with two parameters.

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

Bðn,λk1,k2,⋯,kr;SÞðx, yÞ =Bðn,λk1,k2,⋯,krÞðx + iyÞ− Bðn,λk1,k2,⋯,krÞðx− iyÞ

2i ,

Bðn,λk1,k2,⋯,kr;CÞðx, yÞ =Bðn,λk1,k2,⋯,krÞðx + iyÞ + Bðn,λk1,k2,⋯,krÞðx− iyÞ

2 :

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

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