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

Research Article

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

G. Muhiuddin ,

W. A. Khan,

U. Duran ,

and D. Al-Kadi

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

e^x_λð Þt ≔ 1 + λtð Þ^xλ= 〠^∞

n=0

ð Þx _n,λtⁿ

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→0e^x_λðtÞ = e^xt. Note that e¹_λðtÞ ≔ e_λðtÞ:

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

t

e_λð Þt − 1e^x_λð 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 B_nðxÞ are the familiar Bernoulli polynomials (cf. [1, 3, 4, 6, 8, 11, 12, 14, 16–22])

t

e^t− 1e^xt= 〠^∞

n=0

B_nð Þx tⁿ

n!, ðj j < 2πt Þ: ð4Þ

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

Ei_kð Þ =x 〠^∞

n=1

xⁿ n− 1

ð Þ!n^k, kð ∈ ℤÞ: ð5Þ

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

Eik;λð Þ =x 〠^∞

n=1

ð Þ1 _n,λ n− 1

ð Þ!n^kxⁿ: ð6Þ

Note that Ei_1;λðxÞ = e_λðxÞ − 1:

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

Ei_k;λðlog_λð1 + tÞÞ

e_λð Þt − 1 e^x_λð Þ =t 〠^∞

n=0

B^{ð Þ}_n,^k_λð Þx tⁿ

n!, ð7Þ

log_λð1 + tÞ = 〠^∞

n=1

λⁿ⁻¹ð Þ1 _n,¹

λ

tⁿ

n!,ðλ ∈ ℝÞ, ð8Þ where log_λð1 + 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_λðtÞ as shown below (cf. [8])

e_λðlog_λð1 + tÞÞ = log_λðe_λð1 + tÞÞ = 1 + t: ð9Þ Letting x = 0 in (7), B^ðkÞ_n,λð0Þ ≔ B^ðkÞ_n,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!ðlog_λð1 + tÞÞ^k= 〠^∞

n=k

S_1,λðn, kÞtⁿ

n!, kð ≥ 0Þ, ð10Þ and (cf. [1–27])

1

k!ðe_λð Þt − 1Þ^k= 〠^∞

n=k

S_2,λðn, kÞtⁿ

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

log 1 + tð Þ

ð Þ^k

k! = 〠^∞

n=k

S1ðn, kÞtⁿ

n!ðk≥ 0Þ, ð12Þ

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

k! = 〠^∞

n=k

S₂ðn, kÞtⁿ

n!ðk≥ 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 Ei_{k1,k2,⋯,kr;λ}ðxÞ is defined (cf. [15]) by

Ei_{k1,k2,⋯,kr;λ}ð Þ =x 〠

0<n₁<n₂<⋯<nr

ð Þ1 _n

1,λ⋯ 1ð Þ_n

r,λx^nr n₁− 1

ð Þ!⋯ nð r− 1Þ!n^k1¹⋯ n^kr^r

, ð14Þ

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

2^rEi_{k1,k2,⋯,kr;λ}ðlog_λð1 + tÞÞ e_λð Þ + 1t

ð Þ^r e^x_λð Þ =t 〠^∞

n=0

g^ð_n,^k_λ^1,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;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þ =t 〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð 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 B^ðkn,λ^{1,k2,⋯,krÞ}ð0Þ ≔ B^ðkn,λ^{1,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 e^t− 1

ð Þ^r e^xt= 〠^∞

n=0

B^ðn^{k1,k2,⋯,krÞ}ð 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.

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

d

dxEi_{k1,k2,⋯,kr;λ}ð Þ =x 1

xEi_{k1,k2,⋯,kr−1;λ}ð Þ:x ð18Þ Proof. By (14), we see that

d

dxEi_{k1,k2,⋯,kr;λ}ð Þ =x d

dx 〠

0<n1<n2<⋯<nr

 ð Þ1 _n

1,λ⋯ 1ð Þ_n

r,λx^nr n₁− 1

ð Þ!⋯ nð r− 1Þ!n^k1¹⋯ n^krr

=1

x 〠

0<n₁<n₂<⋯<nr

 ð Þ1 _n1,λ⋯ 1ð Þ_nr,λx^nr n1− 1

ð Þ!⋯ nð r− 1Þ!n^k1¹⋯ n^kr^r−1

=1

xEi_{k1,k2,⋯,kr−1;λ}ð Þ:x

ð19Þ

Theorem 5. The following relationship

B^ðn,^kλ^{1,k2,⋯,krÞ}ð Þ =x 〠ⁿ

j=0

n j !

B^ðn^k−j,λ^1,k2,⋯krÞð Þx _j,λ, ð20Þ

holds for n≥ 0.

Proof. Recall Definition 1 that

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ

n!=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þt

= 〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}tⁿ n!〠^∞

j=0

ð Þx _j,λt^m m!

= 〠^∞

n=0

〠ⁿ

j=0

n j !

B^ð_n−j,λ^{k1,k2,⋯krÞ}ð 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

β^{ð Þ}_n^rðx ;λÞtⁿ

n!= t

e_λð Þt − 1

 r

e^x_λð Þ,t ð22Þ

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

We provide the following theorem.

Theorem 6. For n ≥ r. Then

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þ =x ^n+r〠

m=0

〠

0<n₁<n₂<⋯<nr≤m

 n + r m

!

β^{ð Þ}n+r^r −mðx ;λÞS1:λðm, n_rÞ

× n!r!nr! 1ð Þ_n1,λ⋯ 1ð Þ_nr,λ n + r

ð Þ! nð 1− 1Þ!⋯ nð r− 1Þ!n^k1¹⋯ n^kr^r

: ð23Þ

Proof. Recall from Definition 1 and (10) that

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ

n!= r!e^x_λð Þt e_λð Þt − 1

ð Þ^r 〠

0<n1<n2<⋯<n_r

ð Þ1_n

1,λ⋯ 1ð Þ_n

r,λðlog_λð1 + tÞÞ^nr n₁− 1

ð Þ!⋯ nð _r− 1Þ!n^k₁¹⋯ n^kr^r

= r!e^x_λð Þt e_λð Þt − 1

ð Þ^r 〠

0<n1<n2<⋯<n_r

 ð Þ1_n

1,λ⋯ 1ð Þ_n

r,λn_r! n₁− 1

ð Þ!⋯ nð _r− 1Þ!n^k₁¹⋯ n^kr^r

 〠^∞

m=nr

S_1:λðm, n_rÞt^m m!

=r!

t^r

t^re^x_λð Þt e_λð Þt − 1

ð Þ^r

 

〠^∞

m=nr

 〠

0<n1<n2<⋯<nr≤m

ð Þ1_n

1,λ⋯ 1ð Þ_n

r,λS_1:λðm, n_rÞn_r! n₁− 1

ð Þ!⋯ nð _r− 1Þ!n^k₁¹⋯ n^kr^r

!

t^m m!=r!

t^r〠^∞

l=0

β^{ð Þ}_l^rðx ;λÞt^l l! 〠^∞

m=nr

 〠

0<n1<n2<⋯<nr≤m

ð Þ1_n

1,λ⋯ 1ð Þ_n

r,λS_1:λðm, n_rÞn_r! n₁− 1

ð Þ!⋯ nð _r− 1Þ!n^k₁¹⋯ n^kr^r

!

t^m m!= 〠^∞

n=r

〠ⁿ

m=0

n m !

〠

0<n1<n2<⋯<nr≤m

 r!n_r! 1ð Þ_n

1,λ⋯ 1ð Þ_n

r,λ

n₁− 1

ð Þ!⋯ nð _r− 1Þ!n^k₁¹⋯ n^kr^r

 β^{ð Þ}_n−m^r ðx ;λÞS_1:λðm, n_rÞt^n−r n! ,

ð24Þ

which means the claimed result (23).

Theorem 7. The following formula

B^ð_n,λ^{k1,k2,⋯,krÞ}ðx + yÞ =〠ⁿ

j=0

n j !

ð Þy _j,λB^ð_n−j,λ^{k1,k2,⋯,krÞ}ð Þ,x ð25Þ

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

Proof. In view of Definition 1, we see that

〠^∞

n=0

B^ðn,^kλ^{1,k2,⋯,krÞ}ðx + yÞtⁿ

n!=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x+y_λ ð Þt

=〠^∞

i=0

B^ð_i,λ^{k1,k2,⋯,krÞ}ð Þx tⁱ i!〠^∞

j=0

ð Þy _j,λt^m m!

= 〠^∞

n=0

〠ⁿ

j=0

n j !

ð Þy _m,λB^ðn^k−j,λ^{1,k2,⋯,krÞ}ð Þx

!tⁿ n!, ð26Þ

which implies the desired result (25).

Theorem 8. The following relation

d

dxB^ð_n,λ^{k1,k2,⋯,krÞ}ð Þ=x ⁿ_l=1 n l !

B^ð_n−l,λ^{k1,k2,⋯,krÞ}ð Þx ð Þ−λ^l−1ðl− 1Þ!, ð27Þ

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

Proof. To investigate the derivative property ofB^ðkn,λ^{1,k2,⋯,krÞ}ðxÞ that

〠^∞

n=0

d

dxB^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ

n!= r!Ei_{k1,k2,⋯,kr;λ}ðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r

d dxe^x_λð Þt

= 〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ n!

1

λln 1 +ð λtÞ

= 〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ n!

!

〠^∞

l=1

−1 ð Þ^l+1

l λ^l−1t^l

= 〠^∞

n=0

〠^∞

l=1

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx ð Þ−1^l+1 l λ^l−1t^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^{k1,k2,⋯,krÞ}ð Þ =x 〠ⁿ

m=0

〠^m

l=0

n m !

ð Þx _lS_2,λðm, lÞB^ð_n−m,λ^{k1,k2,⋯,krÞ}, ð29Þ

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

Proof. By means of Definition 1, we attain that

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ð Þx tⁿ

n!=r!Eik1,k2,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þt

=r!Eik1,k2,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r ðe_λð Þt − 1 + 1Þ^x

=r!Eik1,k2,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r 〠^∞

l=0

x l !

e_λð Þt − 1

ð Þ^l

=〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}tⁿ n!〠^∞

l=0

ð Þx_l〠^∞

m=l

S_2,λðm, lÞt^m m!

=〠^∞

n=0

〠ⁿ

m=0

〠^m

l=0

n m !

ð Þx _lS_2,λðm, lÞB^ð_n−m,λ^{k1,k2,⋯,krÞ}

!tⁿ n!,

ð30Þ

where the notation ðxÞ_l is falling factorial that is defined by ðxÞ₀= 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_λð Þt − 1

ð Þ^k

m^kk! e^α_λð Þ =t 〠^∞

n=k

W_m,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 [5]).

We now give a correlation as follows.

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

B^ð_n,λ^{k1,k2,⋯,krÞ}ðxu +αÞ = 〠ⁿ

m=0

〠^m

l=0

n m !

u^lð Þx _lW_u,αðm, ljλÞB^ð_n−m,λ^{k1,k2,⋯,krÞ}: ð32Þ

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

〠^∞

n=0

B^ðn,^kλ^{1,k2,⋯,krÞ}ðxu +αÞtⁿ n!

=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^α_λð Þet ^xu_λ ð Þt

= r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^α_λð Þ et ð^u_λð Þt − 1 + 1Þ^x

=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^α_λð Þt 〠^∞

l=0

x l !

e^u_λð Þt − 1

ð Þ^l

=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r 〠^∞

l=0

u^lð Þx _lðe^u_λð Þt − 1Þ^l l!u^l e^α_λð Þt

= r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r 〠^∞

l=0

u^lð Þx _lðe^u_λð Þt − 1Þ^l l!u^l e^α_λð Þt

= 〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}tⁿ n!〠^∞

n=0

〠ⁿ

l=0

u^lð Þx _lW_u,αðn, ljλÞtⁿ n!

= 〠^∞

n=0

〠ⁿ

m=0

〠^m

l=0

n m !

u^lð Þx _lW_u,αðm, ljλÞB^ð_n−m,λ^{k1,k2,⋯,krÞ}

!tⁿ 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_λt and cosine cos_λt functions by

sin^{ð Þ}_λ^xð Þ =t e^ix_λð Þt − e^−ix_λ ð Þt

2i and cos^{ð Þ}_λ^xð Þ =t e^ix_λð Þ + et ^−ix_λ ð Þt

2 ,

ð34Þ

where i = ffiffiffiffiffiffi p−1

. Note that lim_λ→0sin^ðxÞ_λ ðtÞ = sin xt and lim_λ→0cos^ðxÞ_λ ðtÞ = cos xt. From (34), it is readily that

e^ix_λð Þ = cost ^{ð Þ}_λ^xð Þ + isint ^{ð Þ}_λ^xð Þt : ð35Þ By these functions in (34), the degenerate sine- polynomials S_k,λðx, yÞ and degenerate cosine-polynomials C_k,λðx, yÞ are introduced (cf. [25]) by

〠^∞

n=0

S_k,λðx, yÞtⁿ

n! = e^x_λð Þsint ^{ð Þ}_λ^yð Þ,t ð36Þ

〠^∞

n=0

C_k,λðx, yÞtⁿ

n!= e^x_λð Þcost ^{ð Þ}_λ^yð Þ: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, 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,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x+iy_λ ð Þ =t 〠^∞

n=0

B^ð_n,^k_λ^{1,k2,⋯,krÞ}ðx + iyÞtⁿ n!: ð38Þ

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

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ðx + iyÞ− B^ð_n,λ^{k1,k2,⋯,krÞ}ðx− iyÞ

 

2i

tⁿ n!

= r!Eik1,k2,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þsint ^{ð Þ}_λ^yð Þ,t

ð39Þ

and

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,krÞ}ðx + iyÞ + B^ð_n,λ^{k1,k2,⋯,krÞ}ðx− iyÞ

 

2

tⁿ n!

= r!Eik1,k2,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þcost ^{ð Þ}_λ^yð Þt :

ð40Þ

In view of (39) and (40), we consider the degenerate multi-poly-sine-Bernoulli polynomials B^ðk_n,λ^{1,k2,⋯,kr;SÞ}ðx, yÞ with two parameters and the degenerate multi-poly-cosine- Bernoulli polynomials B^ðk_n,λ^{1,k2,⋯,kr;CÞ}ðx, yÞ with two parame- ters as follows:

〠^∞

n=0

B^ð_n,^k_λ^{1,k2,⋯,kr;SÞ}ðx, yÞtⁿ

n!=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þsint ^{ð Þ}_λ^yð Þ,t ð41Þ

〠^∞

n=0

B^ð_n,λ^{k1,k2,⋯,kr;CÞ}ðx, yÞtⁿ

n!=r!Eik₁,k₂,⋯,kr;λðlog_λð1 + tÞÞ e_λð Þt − 1

ð Þ^r e^x_λð Þcost ^{ð Þ}_λ^yð Þ:t ð42Þ Note that

λ→0limB^ð_n,λ^{k1,k2,⋯,kr;SÞ}ðx, yÞ≔ B^ðn^{k1,k2,⋯,kr;SÞ}ðx, yÞ and

limλ→0B^ð_n,λ^{k1,k2,⋯,kr;CÞ}ðx, yÞ≔ B^ðn^{k1,k2,⋯,kr;CÞ}ðx, yÞ, ð43Þ

which are multi-poly-sine-Bernoulli polynomials B^ðkn^{1,k2,⋯,kr;SÞ}ðx, yÞ and multi-poly-cosine-Bernoulli polyno- mials B^ðkn^{1,k2,⋯,kr;CÞ}ðx, yÞ with two parameters.

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

B^ð_n,λ^{k1,k2,⋯,kr;SÞ}ðx, yÞ =B^ð_n,λ^{k1,k2,⋯,krÞ}ðx + iyÞ− B^ð_n,λ^{k1,k2,⋯,krÞ}ðx− iyÞ

2i ,

B^ð_n,λ^{k1,k2,⋯,kr;CÞ}ðx, yÞ =B^ð_n,λ^{k1,k2,⋯,krÞ}ðx + iyÞ + B^ð_n,λ^{k1,k2,⋯,krÞ}ðx− iyÞ

2 :

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