Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
Research Article
Some identities of q-Euler polynomials under the
symmetric group of degree n
Taekyun Kima,b,∗, Dae San Kimc , Hyuck-In Kwonb , Jong-Jin Seod , D. V. Dolgye
a
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China.
Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea.
c
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.
d
Department of Applied Mathematics, Pukyong National University, Pusan 608-739, S. Korea.
e
Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea.
b
Communicated by R. Saadati
Abstract
In this paper, we investigate some new symmetric identities for the q-Euler polynomials under the
c
symmetric group of degree n which are derived from fermionic p-adic q-integrals on Zp . 2016
All rights
reserved.
Keywords: Identities of symmetry, Carlitz-type q-Euler polynomial, symmetric group of degree n,
fermionic p-adic q-integral.
2010 MSC: 11B68, 11S80, 05A19, 05A30.
1. Introduction
Let p be a fixed prime number such that p ≡ 1 (mod 2). Throughout this paper, Zp , Qp and Cp will
denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic
− 1
closure of Qp . Let q be an indeterminate in Cp such that |1 − q|p < p p−1 . The p-adic norm is normalized
x
as |p|p = p1 and the q-analogue of the number x is defined as [x]q = 1−q
1−q . Note that limq→1 [x]q = x.
As is well known, the Euler numbers are defined by
E0 = 1,
∗
(E + 1)n + En = 2δ0,n ,
(n ∈ N ∪ {0}) ,
Corresponding author
Email addresses: tkkim@kw.ac.kr (Taekyun Kim), dskim@sogang.ac.kr (Dae San Kim), sura@kw.ac.kr (Hyuck-In
Kwon), seo2011@pknu.ac.kr (Jong-Jin Seo), d_dol@mail.ru (D. V. Dolgy)
Received 2015-10-20
T. Kim, et al., J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
1078
with the usual convention about replacing E n by En (see [1–14]).
The Euler polynomials are given by
n X
n n−l
En (x) =
x El = (E + x)n , (n ≥ 0) ,
l
(see [1, 3]) .
l=0
In [4], Kim introduced Carlitz-type q-Euler numbers as follows:
q (qEq + 1)n + En,q = [2]q δ0,n ,
E0,q = 1,
(n ≥ 0) ,
with the usual convention about replacing Eqn by En,q .
The Carlitz-type q-Euler polynomials are also defined as
n n X
n lx
,
En,q (x) = q x Eq + [x]q =
q El,q [x]n−l
q
l
(see [4]) ,
(1.1)
(see [2, 4]) .
(1.2)
l=0
Let C (Zp ) be the space of all Cp -valued continuous functions on Zp . Then, for f ∈ C (Zp ), the fermionic
p-adic q-integral on Zp is defined by Kim as
Z
f (x) dµ−q (x)
(1.3)
I−q (f ) =
Zp
N
pX
−1
1
= lim N
f (x) (−q)x
N →∞ [p ]−q
x=0
N
p −1
1+q X
= lim
f (x) (−q)x ,
N
N →∞ 1 + q p
x=0
(see [4–10]) .
From (1.3), we note that
n−1
n
q I−q (fn ) + (−1)
I−q (f ) = [2]q
n−1
X
(−1)n−1−l f (l) ,
(n ∈ N) ,
(see [4]) .
(1.4)
l=0
The Carlitz-type q-Euler polynomials can be represented by the fermionic p-adic q-integral on Zp as
follows:
Z
En,q (x) =
[x + y]nq dµ−q (y) , (n ≥ 0) , (see [4]) .
(1.5)
Zp
Thus, by (1.5), we get
En,q (x) =
=
n X
n
l=0
n X
l=0
From (1.4), we can easily derive
Z
Z
n
q
[x + 1]q dµ−q (x) +
Zp
l
q lx
Z
[y]lq dµq (y) [x]n−l
q
Zp
n lx
q El,q [x]qn−l ,
l
(see [4]) .
[x]nq dµ−q (x) = [2]q δ0,n ,
(n ∈ N ∪ {0}) .
(1.6)
Zp
The equation (1.6) is equivalent to
qEn.q (1) + En,q = [2]q δ0,n ,
(n ≥ 0) .
(1.7)
The purpose of this paper is to give some new symmetric identities for the Carlitz-type q-Euler polynomials under the symmetric group of degree n which are derived from fermionic p-adic q-integrals on
Zp .
T. Kim, et al., J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
1079
2. Symmetric identities for En,q (x) under Sn
Let w1 , w2 , . . . , wn ∈ N such that w1 ≡ w2 ≡ w3 ≡ · · · ≡ wn ≡ 1 (mod 2). Then, we have


 
Pn−1 Qn−1

 
j=1 wj )x+wn
j=1
i=1 wi kj t
i6=j
q
Q
Q
( n−1 wj )y+( n
j=1
Z
e
dµ−qw1 ···wn−1 (y)
(2.1)
Zp
1
= lim
N →∞ [pN ]q w1 ···wn−1


 
Qn−1
Pn−1 Qn−1
Qn
N −1 (

 
pX
j=1 wj )y+( j=1 wj )x+wn
j=1
i=1 wi kj t
i6=j
q
×
e
(−q w1 ···wn−1 )y
y=0
=
1
lim [2] w1 ···wn−1
2 N →∞ q 
×
wX
n −1
Q
Q
( n−1 wj )(m+wn y)+( n
pN −1
X
j=1
j=1
wj )x+wn
e

 
Q
 n wi kj  t
j=1
i=1
i6=j
q
Pn
m=0 y=0
× (−1)m+y q w1 ···wn−1 (m+wn y) .
Thus, by (2.1), we get
1
[2]qw1 ···wn−1


Pn−1 Qn−1
n−1
wn j=1  i=1 wi kj
l −1
Pn−1
Y wX
ki
i6=j
i=1
(−1)
q

(
Z
×
(2.2)
l=1 kl =0
Qn−1
j=1
wj )y+(
e
 

Pn−1 Qn−1
 

j=1 wj )x+wn
j=1
i=1 wi kj t
i6=j
q
Qn
dµ−qw1 ···wn−1 (y)
Zp
N
−1
n−1
n −1 pX
l −1 wX
Pn−1
Y wX
1
(−1) i=1 ki +m+y
lim
=
2 N →∞
l=1 kl =0 m=0 y=0


Pn−1 Qn−1
Q
Qn
wn j=1  i=1 wi kj +( n−1
j=1 wj )m+( j=1 wj )y
i6=j
×q


 
Q
Q
P
Q
( n−1 wj )(m+wn y)+( n wj )x+wn n−1  n−1 wi kj  t
j=1
j=1
j=1
i=1
i6=j
q
×e
.
As this expression is invariant under any permutation σ ∈ Sn , we have the following theorem.
Theorem 2.1. Let w1 , w2 , . . . , wn ∈ N such that w1 ≡ w2 ≡ · · · ≡ wn ≡ 1 (mod 2). Then, the following
expressions
1
[2]qwσ(1) ···wσ(n−1)


P
Q
n−1
n−1
w
−1
n−1
wσ(n) j=1  i=1 wσ(i) kj
Pn−1
Y σ(l)
X
ki
i6=j
i=1
(−1)
q
l=1
kl =0

j=1
Z
×

 
Pn−1 Qn−1

 
j=1 wj )x+wσ(n)
j=1
i=1 wσ(i) kj t
i6=j
q
Q
Q
( n−1 wσ(j) )y+( n
e
Zp
are the same for any σ ∈ Sn , (n ≥ 1) .
dµqwσ(1) ···wσ(n−1) (y)
T. Kim, et al., J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
1080
Now, we observe that


 



n−1
n
n−1
n−1
Y
Y
X
Y


 
wj  y + 
wj  x + wn
wj  kj  t


j=1

n−1
Y
=
j=1
j=1
 
wj  y + wn x + wn
j=1
n−1
X
j=1
q
i=1
i6=j
(2.3)
q

kj 
wj
.
q w1 ···wn−1
By (2.3), we get

Qn−1
(
Z
j=1

 
Pn−1 Qn−1

 
j=1 wj )x+wn
j=1
i=1 wi kj t
i6=j
q
Qn
wj )y+(
e
dµ−qw1 ···wn−1 (y)
(2.4)
Zp
∞
X
=
n−1
Y

m=0
∞
X
=
m

q
m


m=0
y + wn x + wn
wj 
j=1
n−1
Y

Z
Zp
n−1
X
j=1
m
kj 
wj
q w1 ···wn−1

wj  Em,qw1 ···wn−1 wn x + wn
j=1
dµ−qw1 ···wn−1 (y)
n−1
X
j=1
q
tm
m!

kj  tm
.
wj m!
For m ≥ 0, from (2.4), we have


 m



Z
n−1
n
n−1
n−1
Y
X Y  
 Y 
wj y + 
wj  x + wn
wi  kj  dµ−qw1 ···wn−1 (y)


Zp
j=1
m

n−1
Y
=
j=1
j=1

wj  Em,qw1 ···wn−1 wn x + wn
j=1
n−1
X
j=1
q
i=1
i6=j
(2.5)
q

kj 
,
wj
(n ∈ N) .
Therefore, by Theorem 2.1 and (2.5), we obtain the following theorem.
Theorem 2.2. Let w1 , . . . wn ∈ N be such that w1 ≡ w2 ≡ · · · ≡ wn ≡ 1 (mod 2). For m ≥ 0, the following
expressions


hQ
im
Pn−1 Qn−1
n−1
w
−1


σ(l)
n−1
wσ(n) j=1
Pn−1
Y X
j=1 wσ(j)
i=1 wσ(i) kj
q
ki
i6
=
j
i=1
(−1)
q
[2]qwσ(1) ···wσ(n−1)
l=1 kl =0


m−1
X kj

× Em,qwσ(1) ···wσ(n−1) wσ(n) x + wσ(n)
wσ(j)
j=1
are the same for any σ ∈ Sn .
It is not difficult to show that


n−1
X kj
y + wn x + wn

wj
j=0

= hQ
[wn ]q
n−1
j=1 wj
n−1

(2.6)
q w1 ···wn−1
n−1


X Y  
i 
wi  kj 


q
j=1
i=1
i6=j
q wn
+


Pn−1 Qn−1


wn j=1
i=1 wi kj
i6
=
j
q
[y + wn x]qw1 ···wn−1 .
T. Kim, et al., J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
Thus, by (2.6), we get

m
Z
n−1
X kj
y + wn x + wn

wj
Zp
j=0
q
m X
[wn ]q
m 

=
 hQn−1 i 
l
w
l=0
j
j=1
Z
dµq−w1 ···wn−1 (y)
(2.7)
w1 ···wn−1
m−l 

1081
n−1


n−1
m−l
X  Y  
wi  kj 


j=1
q
i=1
i6=j


Pn−1 Qn−1


lwn j=1
i=1 wi kj
i6
=
j
q
q wn
[y + wn x]lqw1 ···wn−1 dµ−qw1 ···wn−1 (y)
×
Zp

m−l 

 m−l
m n−1
n−1
X
X
Y
[wn ]q
m 



 
=
wi  kj 
 hQn−1 i 


l
w
l=0
j
j=1

lwn
×q
j=1
q
i=1
i6=j
q wn

Pn−1 Qn−1


j=1
i=1 wi kj
i6=j
E
l,q w1 ···wn−1
(wn x) .
From (2.7), we have


Pn−1 Qn−1
n−1
wn j=1  i=1 wi kj
l −1
Pn−1
Y wX
j=1 wj
q
kl
i6=j
l=1
(−1)
q
im
hQ
n−1
[2]qw1 ···wn−1
l=1 kl =0

n
Z
n−1
X kj
y + wn x + wn

×
wj
Zp
j=1
=
m X
m
l=0
l
hQ
n−1
j=1
wj
dµ−qw1 ···wn−1 (y)
q w1 ···wn−1
il
q
[2]qw1 ···wn−1
[wn ]m−l
El,qw1 ···wn−1 (wn x)
q


Pn−1 Qn−1


n−1
(l+1)wn j=1
s −1
Pn−1
Y wX
i=1 wi kj
k
j
i6=j
×
(−1) j=1 q
s=1 ks =0
=
[2]qw1 w2 ···wn−1






n−1
X
j=1
n−1
Y

m−l
 
wi  kj 
i=1
i6=j
q wn
l

1
(2.8)
m n−1
X
m Y 
wj [wn ]m−l
El,qw1 ···wn−1 (wn x)
q
l
l=0
j=1
q
×T̂m,qwn (w1 , w2 , . . . , wn−1 | l) ,
where
T̂m,q (w1 , . . . , wn−1 | l)
=


Pn−1 Qn−1


n−1
s −1 (l+1)
j=1
Y wX
i=1 wi kj
i6
=
j
q
s=1 ks =0






n−1
X
j=1
n−1
Y
i=1
i6=j

m−l
 
wi  kj 
Pn−1
(−1)
j=1
kj
.
q
As this expression is invariant under any permutation in Sn , we have the following theorem.
T. Kim, et al., J. Nonlinear Sci. Appl. 9 (2016), 1077–1082
1082
Theorem 2.3. Let w1 , w2 , . . . , wn ∈ N be such that w1 ≡ w2 ≡ · · · ≡ wn ≡ 1 (mod 2). For m ≥ 0, the
following expressions
1
[2]qwσ(1) wσ(2) ···wσ(n−1)
× El,qwσ(1) ···wσ(n−1)

l
m n−1
X
Y
m−l
m 
wσ(j)  wσ(n) q
l
l=0
j=1
q
wσ(n) x T̂m,qwσ(n) wσ(1) , wσ(2) , . . . , wσ(n−1) | l
are the same for any σ ∈ Sn .
Acknowledgement
Taekyun Kim was appointed as a chair professor at Tianjin polytechnic University by Tianjin city in
China from Aug 2015 to Aug 2019.
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