EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 69-78
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Application of Probabilistic Method on Daehee
Sequences

Chang Liu1,∗, Wuyungaowa1

1 Department of Mathematical Sciences, Inner Mongolia University, Hohhot,
Inner Mongolia, P.R.China

Abstract. In this paper, we investigate some combinatorial sequences based on Daehee and
Changhee numbers and polynomials, then derive their moment representations in use of prob-
abilistic method. We also provide identities related to Daehee numbers, derangement numbers,
Cauchy numbers of the second kind, and Stirling numbers of the first kind.

2010 Mathematics Subject Classifications: 11B68, 60E07, 11B83, 62E15.

Key Words and Phrases: Moment, Probabilistic method, Generating function, Daehee numbers,
Changhee numbers.

1. Introduction and Preliminaries

Throughout this paper, we use the following notations:

N = {1, 2, 3, · · · },Z>0 = {0, 1, 2, · · · }.

Let D
(k)
n,ξ(x) denote the nth twisted Daehee polynomials of order k(∈ N), which are

defined by the generating function[2] to be

(
ln(1 + ξt)

ξt
)k(1 + ξt)x =

∞∑
n=0

D
(k)
n,ξ(x)

tn

n!
. (1)

In special case, when x = 0, D
(k)
n,ξ = D

(k)
n,ξ(0) are called twisted Daehee numbers of

order k. Similarly, D
(k)
n =D

(k)
n,1 are higher-order Daehee numbers, Dn,ξ = D

(1)
n,ξ are twisted

Daehee numbers, and Dn = D
(1)
n,1 are Daehee numbers.

Let D̂
(k)
n,ξ(x) denote the nth twisted Daehee polynomials of the second kind of order

k(∈ N), which are defined by the generating function[2] to be

∗Corresponding author.

Email addresses: changl2013@hotmail.com (C. Liu), wuyungw@163.com (Wuyungaowa)

http://www.ejpam.com 69 c© 2018 EJPAM All rights reserved.



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 70

(
(1 + ξt)ln(1 + ξt)

ξt
)k(1 + ξt)x =

∞∑
n=0

D̂
(k)
n,ξ(x)

tn

n!
. (2)

In special case, when x = 0, D̂
(k)
n,ξ = D̂

(k)
n,ξ(0) are called higher-order twisted Daehee

numbers of the second kind. Similarly, D̂
(k)
n = D̂

(k)
n,1 are higher-order Daehee numbers of

the second kind, D̂n,ξ = D̂
(1)
n,ξ are twisted Daehee numbers of the second kind, and D̂n =

D̂
(1)
n,1 are Daehee numbers of the second kind.

Let Ch
(r)
n (x) denote the nth Changhee polynomials of order r(∈ N), which are defined

by the generating function[1] to be

(
2

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

∞∑
n=0

Ch(r)
n (x)

tn

n!
. (3)

When x = 0, Ch
(r)
n (0) = Ch

(r)
n are called higher-order Changhee numbers.

Let Ĉh
(r)
n (x) denote the nth Changhee polynomials of order r(∈ N) of the second kind,

which are defined by the generating function[1] to be

(
2

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

∞∑
n=0

Ĉh(r)
n (x)

tn

n!
. (4)

When x = 0, Ĉh
(r)
n (0) = Ĉh

(r)
n are called higher-order Changhee numbers of the second

kind.

Remark 1. [see 7] If f and g are exponential generating functions, and

fg = (
∞∑
r=0

arx
r

r!
)(
∞∑
s=0

bsx
s

s!
),

then the coefficients of xn

n! in fg are given by

[
xn

n!
](fg) =

n∑
r=0

(
n

r

)
arbn−r.

Remark 2. Throughout this paper, symbol E denotes the expectation operator defined by

Ef(X) =

∫ +∞

−∞
f(x)p(x)dx,

where random variable X is continuous, whose density function is p(x). Specially, when
f(x) = xn, EXn denotes n-order moment of random variable X.

(i) When r.v u ∼ U [0, 1], Eun = 1
n+1 ,



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 71

(ii) When r.v X ∼ Γ(1, 1), EXn = n!.

Definition 1. The characteristic function of random variable X is defined as

ϕ(t) = EeitX , i2 = −1,−∞ < t <∞. (5)

When the moments of all orders of r.v. X exist, the following relation expression holds
true,

EXn = [
(it)n

n!
]ϕ(t), i2 = −1. (6)

Remark 3. [see 5] If random variable X is distributed as Γ(α, λ), where α, λ > 0, its
characteristic function is

ϕ(t) = EeitX = (1− it

λ
)−α. (7)

Remark 4. X and Y are two random variables, when Cov(X,Y)=0, we have E(XY)=EX·EY,
where

Cov(X,Y ) = E(XY )− E(X)E(Y )

Then we give three lemmas to introduce moment representations of some special com-
binatorial sequences.

Lemma 1. [see 6] Assume that r.v X ∼ Γ(u, 1), with u ∼ U [0, 1] is a random variable that
follows uniform distribution, and X,u are independent respectively, then Cauchy numbers
of the second kind Ĉn, whose generating function is

∞∑
n=0

Ĉn
xn

n!
=

x

(1 + x) ln(1 + x)
, (8)

have the following moment representation,

Ĉn = EXn, n > 0. (9)

Lemma 2. [see 6] Assume that r.v X ∼ Γ(1, 1), then derangement numbers dn =

n!
∑n

k=0
(−1)k

k! have the following moment representation,

dn = E(X − 1)n, n > 0. (10)

Lemma 3. [see 6] Suppose that r.v.s u1, u2, · · · , i.i.d ∼ U [0, 1], r.v.s Γ1,Γ2, · · · , i.i.d ∼
Γ(1, 1), r.v ui and Γj are independent respectively for all i, j. When n, k ≥ 1, Stirling
numbers of the first kind s(n, k) satisfy

s(n, k) = (−1)n−k
(
n

k

)
E(u1Γ1 + u2Γ2 + · · ·+ ukΓk)

n−k. (11)

It is demanded that s(n,0)=s(0,k)=0, s(0,0)=1.



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 72

2. Moment Representations of Daehee and Changhee Sequences

In this section, we use probabilistic method to derive moment representations about
two kinds of higher-order twisted Daehee numbers and polynomials, and two kinds of
higher-order Changhee numbers and polynomials.

Theorem 1. Assume that r.v.s u1, u2, ..., i.i.d ∼ U [0, 1], Γ1,Γ2, ..., i.i.d ∼ Γ(1, 1), and
for all i, j, r.v ui and Γj are independent. When n,m− k ∈ Z>0, k ∈ N, we have

D
(k)
n,ξ = (−ξ)nE(u1Γ1 + u2Γ2 + · · ·+ ukΓk)

n, (12)

D
(k)
m−k,ξ = (−ξ)m−k k

m
E(u1Γ1 + · · ·+ uk−1Γk−1 + Γk)

m−k. (13)

Proof. The generating function of higher-order twisted Daehee numbers is known as

∞∑
n=0

D
(k)
n,ξ

tn

n!
= (

ln(1 + ξt)

ξt
)k, (14)

Taking the coefficients of tn in the left-hand side of Eq.(14), we get

D
(k)
n,ξ

n!
= [tn](

ln(1 + ξt)

ξt
)k = [(−t)n](

∑
i>0

(ξt)i

i+ 1
)k = [(−t)n](

∑
i>0

tiE(ξu)i)k, (15)

(
∑
i>0

tiE(ξu)i)k =
∞∑
n=0

∑
i1+···+ik=n

(E(ξu1)i1) · · · (E(ξuk)
ik)tn

=
∞∑
n=0

1

n!

∑
i1+···+ik=n

(
n

i1, i2, · · · ik

)
(E(ξu1)i1) · · · (E(ξuk)

ik)(i1!) · · · (ik!)tn

=

∞∑
n=0

1

n!

∑
i1+···+ik=n

(
n

i1, i2, · · · ik

)
(E(ξu1)i1) · · · (E(ξuk)

ik)(EΓi11 ) · · · (EΓikk )tn

=

∞∑
n=0

ξn

n!
E[

∑
i1+···+ik=n

(
n

i1, i2, · · · ik

)
(u1Γ1)i1 · · · (ukΓk)ik ]tn

=

∞∑
n=0

ξn

n!
E(u1Γ1 + · · ·+ ukΓk)

ntn. (16)

From Eq.(15) and Eq.(16), we can see that

D
(k)
n,ξ

n!
=

(−ξ)n

n!
E(u1Γ1 + · · ·+ ukΓk)

n,

thus we have
D

(k)
n,ξ = (−ξ)nE(u1Γ1 + · · ·+ ukΓk)

n.



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 73

Eq.(13) can be proved by the following equation[see 6]

E(u1Γ1 + u2Γ2 + · · ·+ ukΓk)
m−k =

k

m
E(u1Γ1 + · · ·+ uk−1Γk−1 + Γk)

m−k (17)

Corollary 1. In theorem 1, when ξ = 1, we obtain the moment representation of higher-
order Daehee numbers[3]

D(k)
n = (−1)nE(u1Γ1 + · · ·+ ukΓk)

n. (18)

Corollary 2. In theorem 1, when k = 1, we obtain the moment representation of twisted
Daehee numbers[2]

Dn,ξ = (−ξ)nE(uΓ)n. (19)

Corollary 3. In theorem1, when ξ = 1, k = 1, we obtain the moment representation of
Daehee numbers[4]

Dn = (−1)nE(uΓ)n. (20)

Theorem 2. Suppose that r.v u ∼ U [0, 1], Γ ∼ Γ(1, 1), r.v u and Γ are independent, then
twisted Daehee numbers of the second kind of order k satisfy

D̂
(k)
n,ξ =

n∑
i=0

(
n

i

)
ξn−i(k)n−iD

(k)
i,ξ . (21)

Proof. The generating function of D̂
(k)
n,ξ is given by

(
ln(1 + ξt)

ξt
(1 + ξt))k =

∞∑
n=0

D̂
(k)
n,ξ

tn

n!
. (22)

From theorem 1, the left-hand side of Eq.(22) can be written as

(
ln(1 + ξt)

ξt
)k(1 + ξt)k

=
∞∑
n=0

(−ξ)nE(u1Γ1 + u2Γ2 + · · ·+ ukΓk)
n t
n

n!

∞∑
n=0

(k)n
(ξt)n

n!

=

∞∑
n=0

n∑
i=0

(
n

i

)
(−ξ)iE(u1Γ1 + u2Γ2 + · · ·+ ukΓk)

i(k)n−iξ
n−i t

n

n!

=

∞∑
n=0

n∑
i=0

(
n

i

)
ξn−i(k)n−iD

(k)
i,ξ

tn

n!
,

where (k)n=k(k − 1) · · · (k − n+ 1). By comparing the coefficients of tn

n! , theorem 2 is
proved.



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 74

Corollary 4. In theorem 2, taking k = 1, n > 1, we obtain the moment form of twisted
Daehee numbers of the second kind,

D̂n,ξ =
(−1)n−1

n
ξnE(uΓ)n, n > 1, D̂0,ξ = 1. (23)

Corollary 5. From corollary 2 and corollary 4, when n > 1,the following relationship
holds true,

−nD̂n,ξ = Dn,ξ, n > 1. (24)

Theorem 3. Under the circumstance of theorem 1, higher-order twisted Daehee polyno-

mials D
(k)
n,ξ(x) have the following moment representation3

D
(k)
n,ξ(x) =

n∑
i=0

(
n

i

)
ξn−i(x)n−iD

(k)
i,ξ . (25)

Proof. Proof of theorem 3 is similar to the one of theorem 2.

Theorem 4. Suppose that r.v.s u1, u2, ..., i.i.d ∼ U [0, 1], Γ1,Γ2, ..., i.i.d ∼ Γ(1, 1),
X ∼ Γ[−x, 1

ξ ], (x < 0, ξ > 0), and r.v ui, Γj and X are independent for all i, j, then

higher-order twisted Daehee polynomials D
(k)
n,ξ(x) satisfy

D
(k)
n,ξ(x) = E[ξ(u1Γ1 + · · ·+ ukΓk) +X]n. (26)

Proof. Replacing t by −it in the generating function of higher-order twisted Daehee
polynomials, and according to remark 3 and 4, we have

∞∑
n=0

D
(k)
n,ξ(x)

(−it)n

n!
= (

ln(1− ξit)
−ξit

)k(1− ξit)−(−x)

=
∞∑
n=0

(−1)nξnE(u1Γ1 + · · ·+ ukΓk)
n (−it)n

n!

∞∑
n=0

EXn (it)n

n!

=

∞∑
n=0

n∑
i=0

(
n

i

)
ξiE(u1Γ1 + · · ·+ ukΓk)

iEXn−i (it)
n

n!

=

∞∑
n=0

E[ξ(u1Γ1 + · · ·+ ukΓk) +X]n
itn

n!
.

By comparing the coefficients of (it)n

n! , we obtain theorem 4.

Theorem 5. Suppose that r.v u ∼ U [0, 1], Γ ∼ Γ(1, 1), r.v u and Γ are independent
respectively, twisted Daehee polynomials of the second kind of order k satisfy

D̂
(k)
n,ξ(x) =

n∑
i=0

(
n

i

)
ξn(−1)iE(u1Γ1 + · · ·+ ukΓk)

i(x+ k)n−i. (27)



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 75

Proof. Proof of theorem 5 is similar to that of theorem 2.

Theorem 6. Suppose that r.v.s u1, u2, ..., i.i.d ∼ U [0, 1], Γ1,Γ2, ..., i.i.d ∼ Γ(1, 1),
X ∼ Γ[−x − k, 1

ξ ], (x < −k, ξ > 0), and r.v ui, Γj and X are independent for all i, j,

then higher-order twisted Daehee polynomials of the second kind D̂
(k)
n,ξ(x) satisfy

D̂
(k)
n,ξ(x) = (−1)nE[ξ(u1Γ1 + · · ·+ ukΓk) +X]n. (28)

Proof. Proof of theorem 6 is similar to the one of theorem 4.

Theorem 7. Assume r.v X ∼ Γ[r, 2], then higher-order Changhee numbers satisfy

Ch(r)
n = (−1)nEXn. (29)

Proof. Replacing t by −it, where i2 = −1 in the generating function of Ch
(r)
n , we have

(
2

2− it
)r =

∞∑
n=0

Ch(r)
n

(−it)n

n!
, (30)

The left-hand side of Eq.(30) can be written as

(1− it

2
)−r = EeitX =

∞∑
n=0

EXn (it)n

n!
, (31)

from Eq.(30) and Eq.(31), we obtain

∞∑
n=0

Ch(r)
n

(−it)n

n!
=
∞∑
n=0

(−1)nEXn (−it)n

n!
.

By comparing the coefficients of (−it)n
n! , theorem 7 is proved.

Theorem 8. Under the circumstance of theorem 7, higher-order Changhee polynomials

of the second kind Ĉh
(r)
n (x) satisfy

Ĉh(r)
n (x) =

n∑
k=0

(
n

k

)
(−1)k(x+ r)n−kEX

k. (32)

Proof. The generating function of Ĉh
(r)
n (x) can be written as

∞∑
n=0

Ĉh(r)
n (x)

tn

n!
= (

2

2 + t
)r(1 + t)x+r

=
∞∑
n=0

(−1)nEXn t
n

n!

∞∑
n=0

(x+ r)n
tn

n!



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 76

=

∞∑
n=0

n∑
k=0

(
n

k

)
(−1)kEXk(x+ r)n−k

tn

n!
.

By comparing the coefficients of tn

n! , theorem 8 is proved.

Corollary 6. In theorem 8, taking x = 0, we obtain the moment form of higher-order
Changhee numbers of the second kind,

Ĉh(r)
n =

n∑
k=0

(
n

k

)
(−1)k(r)n−kEX

k. (33)

Corollary 7. Since the generating function of higher-order Changhee polynomials is

∞∑
n=0

Ch(r)
n (x)

tn

n!
= (

2

t+ 2
)r(1 + t)x. (34)

From theorem 8, we can see that the moment form of Ch
(r)
n (x) is

Ch(r)
n (x) =

n∑
k=0

(
n

k

)
(−1)k(x)n−kEX

k. (35)

3. Identities of Daehee Numbers and Special Combinatorial Sequences

In this section, we use moment forms of special combinatorial sequences, characteristic
function and generating function method to investigate the relationships between Daehee
numbers Dn, Cauchy numbers of the second kind Ĉn, derangement numbers dn, and
Stirling numbers of the first kind, then we obtain combinatorial identities about them.

Theorem 9. Let r.v u ∼ U [0, 1], Γ ∼ Γ(1, 1), X ∼ Γ(u, 1), then Daehee numbers Dn,
Cauchy numbers of the second kind Ĉn, and derangement numbers dn satisfy the following
identity

n∑
k=0

(
n

k

)
(−1)nDkĈn−k =

n∑
k=0

(
n

k

)
dk, n > 0. (36)

Proof. On one hand,

[E(uΓ)− EX]n =
n∑
k=0

(
n

k

)
E(uΓ)k(−EX)n−k

=
n∑
k=0

(
n

k

)
(−1)nE(−uΓ)kEXn−k



C. Liu, Wuyungaowa / Eur. J. Pure Appl. Math, 11 (1) (2018), 69-78 77

=

n∑
k=0

(
n

k

)
(−1)nDkĈn−k, (37)

Write the generating function of the equation above, noting that i2 = −1

∞∑
n=0

[E(uΓ)− EX]n
(it)n

n!
=
∞∑
n=0

n∑
k=0

(
n

k

)
(−1)nDkĈn−k

(it)n

n!

=
∞∑
n=0

Dn
(−it)n

n!

∞∑
n=0

Ĉn
(−it)n

n!

=
ln(1− it)
−it

−it
(1− it) ln(1− it)

= EeitΓ =

∞∑
n=0

EΓn
(it)n

n!
, (38)

On the other hand, by comparing the coefficients of (it)n

n! in Eq.(38), we have

[E(uΓ)− EX]n = EΓn = E(Γ− 1 + 1)n =
n∑
k=0

(
n

k

)
E(Γ− 1)k =

n∑
k=0

(
n

k

)
dk. (39)

From Eq.(37) and Eq.(39), we can get Eq.(36). Theorem 9 is proved.

Theorem 10. Higher-order Deahee numbers D
(k)
n and Stirling numbers of the first kind

satisfy the following relationship

D(k)
n =

s(n+ k, k)(
n+k
k

) . (40)

Proof. From corollary 1 and lemma 3, write generating function of the left-hand side
of Eq.(40),

∞∑
n=0

D(k)
n

tn

n!
=

∞∑
n=0

(−1)nE(u1Γ1 + · · ·+ ukΓk)
n t
n

n!

=
∞∑
n=0

(−1)n
(
n+ k

k

)
E(u1Γ1 + · · ·+ ukΓk)

nk!

tk
tn+k

(n+ k)!

=

∞∑
n=0

s(n+ k, k)(
n+k
k

) tn

n!
.

By comparing the coefficients of tn

n! , we obtain the conclusion.



REFERENCES 78

Acknowledgements

The research is supported by the Natural Science Foundation of China under Grant
11461050 and Natural Science Foundation of Inner Mongolia under Grant 2016MS0104

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