EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5702
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Probabilistic Multiple Poly Bernoulli Polynomials of
the Second Kind

Si Hyeon Lee1,∗, Li Chen2

1 Kwangwoon University, Seoul 139-701, Republic of Korea
2 School of Mathematics, Xi’an University of Finance and Economics, Xi’an 710100, China

Abstract. The purpose of this paper is to introduce the probabilistic multiple poly Bernoulli
polynomials of the second kind under the condition that Y is a random variable. This means
that we will consider the probabilistic extension of the multiple poly Bernoulli polynomials of the
second kind and study to obtain some new results. Furthermore, we investigate their interesting
properties.

2020 Mathematics Subject Classifications: 11B68, 11B73

Key Words and Phrases: Poly Bernoulli polynomials of the second kind, Stirling numbers,
Probabilistic poly-Bernoulli polynomials, Probabilistic Bernoulli polynomials.

1. Introduction

Many years ago Qi-Kim-Kim-Dolgy considered poly Bernoulli polynomials of the sec-
ond kind and multiple poly Bernoulli polynomials of the secon kind in [28]. Recently,
researchers considered probabilistic Stirling numbers, bell numbers, Bernoulli polynomi-
als and Euler polynomials. The aim of this paper is to study probabilistic multiple poly
Bernoulli polynomials of the second kind. Specifically speaking, we assume that Y is a
random variable In section 1, firstly we recall polylogarithm Lik(t), multiple polyloga-
rithm Lik1,...,kr(t). we remind poly Bernoulli polynomials of the second kind and Bernoulli
polynomials of order α. we recall that probabilistic Stirling numbers of the second kind
and Lah numbers. In section 2, we define probabilistic poly Bernoulli polynomials of the

second kind b
(k)
n,Y and probabilistic multiple poly Bernoulli polynomials of the second kind

b
(k1,...,kr)
n,Y associated with Y . In Theorem 2.1, we derive an expression for nb

(k)
n−1,Y (x).

When Y ∼ Γ(1, 1) in Theorem 2.2, we get an expression for b
(k)
n,Y (x) as sum of the prod-

ucts. When Y is the Bernoulli random variable, in Theorem 2.3 we get expression for

b
(k)
n,Y (x). In Theorem 2.4, we obtain an expression for b

(k1,...,kr)
n,Y (x). In Theorem 2.5, we

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5702

Email addresses: ugug11@naver.com (S. H. Lee), chenli 0928@xaufe.edu.cn (L. Chen)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 2 of 13

find an expression of
b
(k1,...,kr)
n+1,Y (x+1)−b

k1,...,kr
n+1,Y (x)

n+1 . In Theorem 2.6, we derive an expression of

b
(k1,...,kr)
n,Y (x) between probabilistic multi-poly-Bernoulli polynomials, Bernoulli polynomi-
als of the second kind of order r and probabilistic the Stirling numbers of the second kind.

In Theorem 2.7 we get an expression of b
(k1,...,kr)
n,Y (x+ y). Now we recall that

The Bernoulli polynomials of the second kind are defined by

t

log(1 + t)
(1 + t)x =

∞∑
n=0

bn(x)
tn

n!
, (see[12], [13], [25], [29]). (1)

The Bernoulli polynomials of the second kind with order r are defined by the generating
function (

t

log(1 + t)

)r

(1 + t)x =
∞∑
n=0

brn(x)
tn

n!
, (r ∈ Z), (see[17]). (2)

It is well known that

t(1 + t)x−1

log(1 + t)
=

∞∑
n=0

B(n)
n (x)

tn

n!
, (see[9], [22], [5].[7]). (3)

where the Bernoulli polynomials of order α which are given by(
t

et − 1

)α

ext =
∞∑
n=0

Bα
n (x)

tn

n!
, (see[9], [22], [5], [29], [8].[7]). (4)

From (1) and (3), we note that

bn(x) = B(n)
n (x+ 1), (n ≥ 0), (see[9], [22]). (5)

For k ∈ Z, the polylogarithm function is defined by

Lik(x) =
∞∑
n=1

xn

nk
, (see[4], [13], [9], [22], [11], [16], [21]). (6)

The poly Bernoulli polynomials of the second kind are defined by

Lik(1− e−t)

log(1 + t)
(1 + t)x =

∞∑
n=0

b(k)n (x)
tn

n!
, (see[9], [23]). (7)

For k1, k2, . . . , kr ∈ Z, the multiple polylogarithm is defined by

Lik1,...,kr(z) =
∑

0<n1<···<nr

znr

nk1
1 · · ·nkr

r

, (|t| < 1), (see[9], [21], [18], [24], [25], [27], [28], [26]).

(8)



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 3 of 13

By (8), we note that

d

dt
Lik,1(t) =

1

1− t
Lik(t), (see[9], [21], [24], [25], [27], [28]). (9)

It is obvious that

Li1,1(t) =

∫ t

0

d

dt
Li1,1(t)dt =

∫ t

0

1

1− t
(−log(1− t))dt =

1

2!
(−log(1− t))2. (10)

Continuing this process in (10), we have

Li1,...,1(t) =
1

r!
(−log(1− t))r, (r ∈ N), (see[3], [9], [21], [24], [25], [27], [28]). (11)

Throught this paper, we assume that Y is a random variable such that the moment
generating function of Y given by

E[eY t] =
∞∑
n=0

E[Y n]
tn

n!
, (|t| < r), (see[15], [12], [13], [19], [10], [30]). (12)

exists for some r > 0. Let (Yi)i≥1 be a sequence of mutually independent copies of random
variable Y , and let Sn = Y1 + · · · + Yn, (n ≥ 1), with S0 = 0. A continuous random
variable Y whose density function is given by

f(y) =

{
βe−βy (βy)α−1

Γ(α) , if y > 0,

0, if y ≤ 0,
(see[13], [16], [20]), (13)

for some α, β > 0 is said to be the gamma random variable with parameters α, β, which
is denoted by Y ∼ Γ(α, β).

The Stirling number of the second kind are defined by

xn =

n∑
k=0

S2(n, k)(x)k, (see[2], [19], [6], [14], [29]). (14)

From (14), we also derive the generating function as follows.

1

k!
(et − 1)k =

∞∑
n=k

S2(n, k)
tn

n!
, (see[2], [19], [6], [14]). (15)

It is well known that the Lah numbers are defined by

1

k!

(
t

1− t

)k

=

∞∑
n=k

L(n, k)
tn

n!
, (r ≥ 0), (see[13], [6]). (16)

The probabilistic Stirling number of the second kind associated with Y are defined by

1

k!
(E[eY t]− 1)k =

∞∑
n=k

{
n
k

}
Y

tn

n!
, (see[1], [13], [19]). (17)



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 4 of 13

In [13]. Kim also considered the probabilistic multi-poly-Bernoulli polynomials associated
with Y by

Lik1,...,kr
(
1− E[e−Y t]

)
(1− E[e−Y t])r

(
E[e−Y t]

)x
=

∞∑
n=0

B
(k1,...,kr)
n,Y (x)

tn

n!
, (see[13]). (18)

2. probabilistic poly and multiple poly Bernoulli polynomials of the
second kind

In this section, we consider probabilistic poly Bernoulli polynomials of the second kind
associated with Y which are given by

Lik(1− E[e−Y t])

log(1 + t)
(1 + t)x =

∞∑
n=0

b
(k)
n,Y (x)

tn

n!
. (19)

When x = 0, b
(k)
n,Y (0) = b

(k)
n,Y are called probabilistic multiple poly Bernoulli numbers of

the second kind.
From (19), we have
Proof. Theorem 1.

t
∞∑
n=0

b
(k)
n,Y (x)

tn

n!
=

t

log(1 + t)
(1 + t)xLik(1− E[e−Y t]) (20)

=
∞∑
l=0

B
(l)
l (x+ 1)

tl

l!

∞∑
m=1

(1− E[e−Y t])m

mk

=
∞∑
l=0

B
(l)
l (x+ 1)

tl

l!

∞∑
m=1

(−1)mm!

mk

∞∑
j=m

(−1)j
{
j

m

}
Y

tj

j!

=

∞∑
l=0

B
(l)
l (x+ 1)

tl

l!

∞∑
j=1

j∑
m=1

(−1)m+jm!

mk

{
j

m

}
Y

tj

j!

=

∞∑
n=1

n∑
j=1

j∑
m=1

(−1)m+jm!

mk

(
n

j

){
j

m

}
Y

B
(n−j)
n−j (x+ 1)

tn

n!
.

On the other hand, in (20)

t

∞∑
n=0

b
(k)
n,Y (x)

tn

n!
=

∞∑
n=1

nb
(k)
n−1,Y (x)

tn

n!
. (21)

Thus, by comparing the coefficients on both sides of (20) and (21), we have the following
theorem.



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 5 of 13

Theorem 1. For n ≥ 1, we have

nb
(k)
n−1,Y (x) =

n∑
j=1

j∑
m=1

(−1)m+jm!

mk

(
n

j

){
j

m

}
Y

B
(n−j)
n−j (x+ 1).

Let Y ∼ Γ(1, 1), then we note that

E[eY t] =
1

1− t
. (22)

From (19) and (22), we have
Proof. Theorem 2.

∞∑
n=0

b
(k)
n,Y (x)

tn

n!
=

t(1 + t)x

tlog(1 + t)

∑
m=1

(1− 1
1−t)

m

mk
(23)

=
1

t

∞∑
i=0

bi(x)
ti

i!

∞∑
m=1

(−1)mm!

mk

1

m!

(
t

1− t

)m

=
1

t

∞∑
i=0

bi(x)
ti

i!

∞∑
m=1

(−1)mm!

mk

∞∑
l=m

L(l,m)
tl

l!

=
1

t

∞∑
i=0

bi(x)
ti

i!

∞∑
l=1

l∑
m=1

(−1)mm!

mk
L(l,m)

tl

l!

=

∞∑
n=1

n∑
l=1

l∑
m=1

(
n

l

)
(−1)mm!

mk
L(l,m)bn−l(x)

tn−1

n!

=

∞∑
n=0

n+1∑
l=1

l∑
m=1

(
n+ 1

l

)
(−1)mm!

mk
L(l,m)

bn−l+1(x)

n+ 1

tn

n!
.

Thus, we have the following theorem.

Theorem 2. Let Y ∼ Γ(1, 1). For n ≥ 0, we have

b
(k)
n,Y (x) =

n+1∑
l=1

l∑
m=1

(
n+ 1

l

)
(−1)mm!

mk
L(l,m)

bn−l+1(x)

n+ 1
.

Let Y be the Bernoulli random variable with probability of success p. Then we have

E[eY t] = p(et − 1) + 1. (24)

By (19) and (24), we have



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 6 of 13

Proof. Theorem 3.

∞∑
n=0

b
(k)
n,Y

tn

n!
=

(1 + t)x

log(1 + t)

∞∑
m=1

(1− E[e−Y t])m

mk
(25)

=
(1 + t)x

log(1 + t)

∞∑
m=1

(−1)mpm(et − 1)m

mk

=
t(1 + t)x

tlog(1 + t)

∞∑
m=1

(−1)mpmm!

mk

(et − 1)m

m!

=
1

t

∞∑
l=0

bl(x)
tl

l!

∞∑
m=1

(−1)mpmm!

mk

∞∑
i=m

S2(i,m)
ti

i!

=
1

t

∞∑
l=0

bl(x)
tl

l!

∞∑
i=1

i∑
m=1

(−1)mpmm!

mk
S2(i,m)

ti

i!

=

∞∑
n=1

n∑
i=1

i∑
m=1

(
n

i

)
(−1)mpmm!

mk
S2(i,m)bn−i(x)

tn−1

n!

=
∞∑
n=0

n+1∑
i=1

i∑
m=1

(
n+ 1

i

)
(−1)mpmm!

mk
S2(i,m)

bn−i+1(x)

n+ 1

tn

n!
.

Thus, we have the following theorem.

Theorem 3. Let Y be the Bernoulli random variable with probability of success p. For
n ≥ 0, we have

b
(k)
n,Y =

n+1∑
i=1

i∑
m=1

(
n+ 1

i

)
(−1)mpmm!

mk
S2(i,m)

bn−i+1(x)

n+ 1
.

Now, we consider probabilistic multiple poly Bernoulli polynomials of the second kind
which are given by

r!Lik1,...,kr(1− E[e−Y t])

(log(1 + t))r
(1 + t)x =

∞∑
n=0

b
(k1,...,kr)
n,Y (x)

tn

n!
. (26)

When k1 = · · · = kr = 1 and Y = 1, we note that

∞∑
n=0

b
(1,...,1)
n,Y (x) =

(
t

log(1 + t)

)r

(1 + t)x =
∞∑
n=0

brn(x)
tn

n!
. (27)

From (26), we have



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 7 of 13

Proof. Theorem 4.

∞∑
n=0

b
(k1,...,kr)
n,Y (x)

tn

n!
=

r!(1 + t)x

(log(1 + t))r
Lik1,...,kr(1− E[e−Y t]) (28)

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr

(
1− E[e−Y t]

)mr

mk1
1 · · ·mkr

r

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

1

mk1
1 · · ·mkr−1

r−1

∞∑
mr=mr−1+1

(
1− E[e−Y t]

)mr

mkr
r

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

×
∞∑

mr=0

(−1)mr+1(mr + 1)!

(mr +mr−1 + 1)kr

(
E[e−Y t − 1]

)mr+1

(mr + 1)!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

×
∞∑

mr=0

(−1)mr+1(mr + 1)!

(mr +mr−1 + 1)kr

∞∑
l=mr+1

{
l

mr + 1

}
Y

(−1)ltl

l!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

×
∞∑
l=1

l−1∑
mr=0

(−1)m
r+l+1(mr + 1)!

(mr +mr−1 + 1)kr

{
l

mr + 1

}
Y

tl

l!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1−j

r−1

×
∞∑
l=1

l−1∑
mr=0

(−1)m
r+l+j+1(mr + 1)!

{
l

mr + 1

}
tl

l!

∞∑
j=0

(
kr + j − 1

j

)
(mr + 1)−kr−j t

j

j!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1−j

r−1

×
∞∑
l=1

l−1∑
mr=0

∞∑
j=0

(−1)mr+l+j+1(mr + 1)!(mr + 1)−kr−j

{
l

mr + 1

}
Y

(
kr + j − 1

j

)
tl

l!

=
r

log(1 + t)

∞∑
l=1

l−1∑
mr=0

∞∑
j=0

(−1)mr+l+j+1(mr + 1)!(mr + 1)−kr−j

{
l

mr + 1

}
Y

(
kr + j − 1

j

)
tl

l!



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 8 of 13

×
∞∑
i=0

b
(k1,...,kr−1−j)
i,Y (x)

ti

i!

=
r

Li1(−t)

∞∑
n=1

n−1∑
l=0

l−1∑
mr=0

∞∑
j=0

(−1)mr+l+j

(
n

l + 1

)(
kr + j − 1

j

){
l + 1

mr + 1

}
Y

× (mr + 1)!(mr + 1)−kr−jb
(k1,...,kr−1−j)
n−l−1,Y (x)

tn

n!
.

Therefore, by (28), we have the following theorem.

Theorem 4. For n ≥ 0, t ̸= 0, we have

b
(k1,...,kr)
n,Y (x) =

r

Lik(−t)

n−1∑
l=0

l−1∑
mr=0

∞∑
j=0

(−1)mr+l+j

(
n

l + 1

)(
kr + j − 1

j

){
l + 1

mr + 1

}
Y

× (mr + 1)!(mr + 1)−kr−jb
(k1,...,kr−1−j)
n−l−1,Y (x).

From (26), we obtain
Proof. Theorem 5.

1

t

( ∞∑
n=0

b
(k1,...,kr)
n,Y (x+ 1)

tn

n!
−

∞∑
n=0

b
(k1,...,kr)
n,Y (x)

tn

n!

)
=

r!Lik1,...,kr
(
1− E[e−Y t]

)
(log(1 + t))r

(1 + t)x

(29)

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr

(
1− E[e−Y t]

)mr

mk1
1 · · ·mkr

r

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

1

mk1
1 · · ·mkr−1

r−1

∑
mr=mr−1+1

(
1− E[e−Y t]

)mr

mkr
r

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[eY t]

)mr−1

mk1
1 · · ·mkr−1

r−1

∞∑
mr=1

(−1)mrmr!

(mr +mr−1)kr

(
E[e−Y t]− 1

)mr

mr!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[eY t]

)mr−1

mk1
1 · · ·mkr−1

r−1

∞∑
mr=1

(−1)mr+lmr!

(mr +mr−1)kr

∞∑
l=mr

{
l

mr

}
Y

tl

l!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[eY t]

)mr−1

mk1
1 · · ·mkr−1

r−1

∞∑
l=1

l∑
mr=1

(−1)mr+lmr!

(mr +mr−1)kr

{
l

mr

}
Y

tl

l!

=
r!(1 + t)x

(log(1 + t))r

∑
0<m1<···<mr−1

(
1− E[eY t]

)mr−1

mk1
1 · · ·mkr−1−j

r−1

×
∞∑
l=1

l∑
mr=1

∞∑
j=0

(
kr + j − 1

j

)
(−1)mr+l+jm−kr−j

r mr!

{
l

mr

}
Y

tl

l!



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 9 of 13

=
r

log(1 + t)

∞∑
i=0

b
(k1,...,kr−1−j)
i,Y (x)

ti

i!

∞∑
l=1

l∑
mr=1

∞∑
j=0

(
kr + j − 1

j

)
(−1)mr+l+jm−kr−j

r mr!

{
l

mr

}
Y

tl

l!

=
r

Li1(−t)

∞∑
n=1

n∑
l=1

l∑
mr=1

∞∑
j=0

(
kr + j − 1

j

)(
n

l

)
(−1)mr+l+jm−kr−j

r mr!

{
l

mr

}
Y

b
(k1,...,kr−1−j)
n−l,Y (x)

tn

n!
.

On the other hand in (29)

1

t

( ∞∑
n=0

b
(k1,...,kr)
n,Y (x+ 1)

tn

n!
−

∞∑
n=0

b
(k1,...,kr)
n,Y (x)

tn

n!

)
=

∞∑
n=0

(
b
k1,...,kr
n+1,Y (x+ 1)− bk1,...,krn+1,Y (x)

)
n+ 1

tn

n!
.

(30)

By comparing the coefficients on both sides of (29) and (30), we have the following
theorem.

Theorem 5. For n ≥ 1, t ̸= 0, we have(
b
(k1,...,kr)
n+1,Y (x+ 1)− b

(k1,...,kr)
n+1,Y (x)

)
n+ 1

=
r

Li1(−t)

n∑
l=1

l∑
mr=1

∞∑
j=0

(
kr + j − 1

j

)(
n

l

)
(−1)mr+l+jm−kr−j

r mr!

{
l

mr

}
Y

b
(k1,...,kr−1−j)
n−l,Y (x).

From (26), we have
Proof. Theorem 6.

∞∑
n=0

b
(k1,...,kr)
n,Y (x)

tn

n!
=

r!Lik1,...,kr
(
1− E[e−Y t]

)
(log(1 + t))r

(1 + t)x (31)

=
trr!(1 + t)x

tr((log(1 + t))r
Lik1,...,kr

(
1− E[e−Y t]

)
(1− E[e−Y t])r

(
1− E[e−Y t]

)r
=

(r!)2

tr

∞∑
m=0

B
(k1,...,kr)
m,Y

tm

m!

∞∑
l=r

{
l

r

}
Y

(−1)l
tl

l!

∞∑
i=0

bri (x)
ti

i!

=
(r!)2

tr

∞∑
j=0

j∑
m=0

(
j

m

)
B

(k1,...,kr)
m,Y brj−m(x)

tj

j!

∞∑
l=r

{
l

r

}
Y

(−1)l
tl

l!

=
(r!)2

tr

∞∑
n=r

n∑
l=0

n−l∑
m=0

(
n− l

m

)(
n

l

)
(−1)lB

(k1,...,kr)
m,Y b

(r)
n−l−m(x)

{
l

r

}
Y

tn

n!

= (r!)2
∞∑
n=0

n+r∑
l=0

n+r−l∑
m=0

(
j

m

)(
n+ r − l

l

)
(−1)l

B
(k1,...,kr)
m,Y b

(r)
n+r−l−m(x)

(n+ r)r

{
l

r

}
Y

tn

n!
.



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 10 of 13

Therefore, by (31), we have the following theorem.

Theorem 6. For n ≥ 0, we have

b
(k1,...,kr)
n,Y (x) = (r!)2

n+r∑
l=0

r−l∑
m=0

(
j

m

)(
n+ r

j

)
(−1)l

B
(k1,...,kr)
m,Y b

(r)
n+r−l−m(x)

(n)r

{
l

r

}
Y

.

From (26), we have
Proof. Theorem 7.

∞∑
n=0

b
(k1,...,kr)
n,Y (x+ y)

tn

n!
=

r!Lik1,...,kr(1− E[e−Y t])

(log(1 + t))r
(1 + t)x+y (32)

=
r!(1 + t)x

(log(1 + t))r

∞∑
l=0

(y)l
tl

l!

∑
0<m1<···<mr−1

1

mk1
1 · · ·mkr−1

r−1

∞∑
mr=mr−1+1

(
1− E[e−Y t]

)mr

mkr
r

=
r!(1 + t)x

(log(1 + t))r

∞∑
l=0

(y)l
tl

l!

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

×
∞∑

mr=1

(−1)mrmr!

(mr +mr−1)k

(
E[e−Y t]− 1

)mr

mr!

=
r!(1 + t)x

(log(1 + t))r

∞∑
l=0

(y)l
tl

l!

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

×
∞∑

mr=1

(−1)mrmr!

(mr +mr−1)k

∞∑
i=mr

{
i

mr

}
Y

(−1)i
ti

i!

=
r!(1 + t)x

(log(1 + t))r

∞∑
l=0

(y)l
tl

l!

∑
0<m1<···<mr−1

(
1− E[e−Y t]

)mr−1

mk1
1 · · ·mkr−1

r−1

∞∑
i=1

i∑
mr=1

(−1)mr+imr!
{

i
mr

}
(mr +mr−1)k

ti

i!

=
∞∑
i=1

i∑
mr=1

∞∑
j=0

(
mr + j − 1

j

)
(−1)mr+i−jm−kr−j

r mr!

{
i

mr

}
Y

ti

i!

× r!(1 + t)x

(log(1 + t))r
Lik1,...,kr−1−j

(
1− E[e−Y t]

) ∞∑
l=0

(y)l
tl

l!

=
r

log(1 + t)

∞∑
i=1

i∑
mr=1

∞∑
j=0

(
mr + j − 1

j

)
(−1)mr+i−jm−kr−j

r mr!

{
i

mr

}
Y

ti

i!

×
∞∑
s=0

b
(k1,...,kr−1−j)
s,Y (x)

ts

s!

∞∑
l=0

(y)l
tl

l!



S. H. Lee, L. Chen / Eur. J. Pure Appl. Math, 18 (1) (2025), 5702 11 of 13

=
r

log(1 + t)

∞∑
u=0

u∑
l=0

(
u

l

)
b
(k1,...,kr−1−j)
u−l,Y (x)

tu

u!

×
∞∑
i=1

i∑
mr=1

∞∑
j=0

(
mr + j − 1

j

)
(−1)mr+i+jm−kr−j

r mr!

{
i

mr

}
Y

ti

i!

=
r

Li1(−t)

∞∑
n=1

n∑
i=1

i∑
mr=1

∞∑
j=0

(
n− i

l

)(
n

i

)(
mr + j − 1

j

)
(−1)mr+i+jm−kr−j

r mr!

×
{

i

mr

}
Y

b
(k1,...,kr−1−j)
n−i−l,Y (x)

tn

n!
.

By comparing the coefficients on both sides of (32), we have following theorem.

Theorem 7. For n ≥ 1, we have

b
(k1,...,kr)
n,Y (x+ y)

=
r

Li1(−t)

n∑
i=1

i∑
mr=1

∞∑
j=0

(
n− i

l

)(
n

i

)(
mr + j − 1

j

)
(−1)mr+i+jm−kr−j

r mr!

×
{

i

mr

}
Y

b
(k1,...,kr−1−j)
n−i−l,Y (x).

3. Conclusion

An increasing number of scholars have studied many special polynomials and numbers
and have applied the properties of these polynomials in probability theory. In view of
this, we explored the properties of probabilistic multiple poly Bernoulli polynomials of the
second kind. That implies that we have conducted an in-depth study of these polynomials
and numbers. We obtained several interesting results and formula when Y is the Bernoulli
random variable and gamma random variable. In the future, we will consider more valuable
polynomials and numbers and get some significant results.

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