Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 417452, 12 pages
doi:10.1155/2010/417452
Research Article
A Note on Nörlund-Type Twisted q-Euler
Polynomials and Numbers of Higher Order
Associated with Fermionic Invariant q-Integrals
Leechae Jang
Department of Mathematics and Computer Science, KonKuk University, Seol 143-701, Republic of Korea
Correspondence should be addressed to Leechae Jang, leechae.jang@kku.ac.kr
Received 4 April 2010; Accepted 20 June 2010
Academic Editor: Vijay Gupta
Copyright q 2010 Leechae Jang. 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.
We define multiple Nörlund-type twisted q-Euler polynomials and numbers and give interpolation functions of multiple Nörlund-type twisted q-Euler polynomials at negative integers.
Furthermore, we investigate some identities related to these polynomials and interpolation
functions.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will be,
respectively, the ring of p-adic rational integers, the field of p-adic rational numbers, and
the p-adic completion of the algebraic closure of Qp . The p-adic absolute value in Cp is
normalized so that |p|p 1/p. When one talks of q-extension, q is variously considered as
an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, one normally
assumes |q| < 1. If q ∈ Cp , one normally assumes |1 − q|p < p1/1−p so that qx expx log q
for each x ∈ Zp . We use the notation
xq 1 − qx
,
1−q
compared with 1–22, for all x ∈ Zp .
x−q
x
1 − −q
,
1q
1.1
2
Journal of Inequalities and Applications
For a fixed odd positive integer d with p, d 1, set
Z/dpn Z,
X Xd lim
←
−
X1 Zp ,
n
X∗ a dpZp ,
1.2
0<a<dp
a,p1
a dpn Zp x ∈ X | x ≡ a moddpn ,
where a ∈ Z lies in 0 ≤ a < dpn . For any n ∈ N,
qa
μq a dpn Zp n dp q
1.3
is known to be a distribution on Zp , compared with 1–22.
The q-factorial is defined as nq ! nq n − 1q · · · 2q 1q , and the Gaussian binomial
coefficient is also defined by
nq n − 1q · · · n − k 1q
nq !
n
k q n − kq !kq !
kq !
1.4
nn − 1 · · · n − k 1
n!
n
n
.
k
q→1 k
k!
−
k!k!
n
q
1.5
see 7, 8.
Note that
lim
From 1.4, we note that
n1
n
n
n
k n
n−1
q
q
k
k
−
1
k q
k
k
−
1
q
q
q
q
1.6
see 7, 8.
The q-binomial formulae are known as
b:q
n
1 − b 1 − bq · · · 1 − bq
n−1
n
n
i0
1
1
b : q n 1 − b 1 − bq · · · 1 − bqn−1
i
q
i
2 −1i bi ,
q
1.7
∞
ni−1
i0
i
bi .
q
Journal of Inequalities and Applications
3
The Euler number En and polynomials En x are defined by the generating function
in the complex number field as
∞
2
tn
E
n
et 1 n0 n!
|t| < π,
1.8
∞
tn
2
xt
e
E
x
n
n!
et 1
n0
|t| < π.
The nth q-Euler numbers En,q and the nth q-Euler polynomials En,q x attached to q are
defined by the exponential generating functions as
∞
∞
tn
En,q ,
Fq t 2 −1k ekq t n!
n0
k0
1.9
∞
∞
tn
Fq t, x 2 −1k ekxq t En,q x .
n!
n0
k0
r
r
The nth Euler numbers En of higher order and the nth Euler polynomials En x of higher
order attached to q are defined by the exponential generating functions as
F r t ∞
n
2r
r t
,
E
n
r
n!
1 et k0
1.10
∞
tn
2r
r
xt
,
e
E
x
n
r
n!
1 et k0
r
∞
1 et
tn
−r
−r
xt
F t, x .
e
E
x
n
2r
n!
k0
F r t, x 1.11
1.11 − 1
r
r
Kim 7 defined the nth q-Euler numbers En,q of higher order, the Euler polynomials
En,q x of higher order, and the nth Nörlund-type q-Euler polynomials of higher order which
are defined by the exponential generating functions as
r
Fq t
2
r
∞
−1
mr−1
m
m
m0
r
Fq t, x 2r
∞
−1m
m0
−r
Fq
t, x ∞
n0
compared with 6–16, 18–21.
q
mr−1
m
emq t emxq t q
−r
En,q x
tn
,
n!
∞
n
r t
En,q ,
n!
n0
∞
tn
r
En,q x ,
n!
n0
1.12
4
Journal of Inequalities and Applications
We say that f is uniformly differentiable function at a point a ∈ Zp and denote this
property by f ∈ UDZp if the difference quotients
fx − f y
Ff x, y x−y
1.13
have a limit l f a as x, y → a, a, compared with 1–22 23-24. Note that the bosonic
p-adic q-integral of a function f ∈ UDZp was defined by
Iq f p −1
1 fxdμq x lim n fxqx
n
→
∞
p
Zp
q x0
n
1.14
and that the fermionic p-adic q-integral was defined by
I−q f p −1
x
1 fxdμ−q x lim n fx −q
n→∞ p
Zp
−q x0
n
1.15
see 1–22 23-24 . In 1.15, when q → 1, we can obtain
L−1 f1 L−1 f 2f0,
1.16
where f1 x fx 1. If we take fx etx , then we obtain
I−1 e
tx
Zp
etx dμ−1 x et
2
.
1
1.17
In this paper, we define multiple Nörlund-type twisted q-Euler polynomials and give
interpolation functions of multiple Nörlund-type twisted q-Euler polynomials at negative
integers. Furthermore, we investigate some identities related to these polynomials and
interpolation functions.
Journal of Inequalities and Applications
5
2. Nörlund-Type Twisted q-Euler Numbers and
Polynomials of Higher Order
In this section, we assume that q ∈ Cp with |1 − q|p < 1. Let Cp∞ n≥1 Cpn limn → ∞ Cpn be
n
the locally constant space, when Cpn {ξ ∈ X | ξp 1} is the cyclic group of order pn . Let
ξ ∈ Cp∞ . We define the twisted q-Euler polynomials see 1–5, 18–21 as follows:
r
Em,q,ξ x ···
Zp
Zp
x x1 · · · xr nq ξx1 ···xr dμ−1 x1 · · · dμ−1 xr n n
−1l qlx
n
l
1 − q l0
2r
2
r
∞ mr−1
m
m0
r
r
1
1 ql ξ l
r
2.1
−1m m xnq,ξm ,
where aq,ξ 1 − qa ξ/1 − q. Let Fq,ξ t, x Fq,ξ t, x 2r
∞
n0
∞ mr−1
m0
r
m
r
En,q,ξ xtn /n!. Then we have
−1m emxq,ξm t .
2.2
r
In this special case x 0, En,q,ξ 0 En,q,ξ are called the twisted q-Euler numbers of
order r. In the sense of the twisted in 1.11 − 1, we consider the Nörlund-type twisted q-Euler
polynomials as follows:
r
−r
Gq,ξ t, x Fq,ξ t, x ∞
r tn
1
r
−r
En,q,ξ x .
emxq,ξm t r
2 m0 m
n!
n0
2.3
By 2.3, we have
−r
En,q,ξ x
r 1
r
r
m xnq,ξm .
m
2 m0
2.4
From 2.1 and 2.4, we can obtain the following theorem.
Theorem 2.1. For r ∈ N, n ≥ 0, and ε ∈ Tp , let
2r
∞ mr−1
m0
m
−ξm emxq,ξm t ∞
tn
r
En,q,ξ x .
n!
n0
2.5
6
Journal of Inequalities and Applications
Then
r
En,q,ξ x
−r
En,q,ξ x
−r
n n
2r
l lx
1
1 ql ξ l
r
−1 q
n
1 − q l0 l
∞ mr−1
2r
−1m m xnq,ξm ,
m
m0
n n
1
q 1qξ
n
2r 1 − q l0 l
r 1
r
r
m xnq,ξm .
2 m0 m
lx
l l
2.6
r
−r
En,q,ξ 0 En,q,ξ are called the twisted q-Euler numbers of higher order. For h ∈ Z,
r ∈ N, let us define the twisted q-Euler polynomials of higher order as follows:
h,r
En,q,ξ x
Zp
···
Zp
q
r
j1 h−jxj
ξ
r
j1
xj
x x1 · · · xr nq dμ−1 x1 · · · dμ−1 xr .
2.7
Then
2r
h,r
En,q,ξ x 1−q
2r
2
m0
h,r
h,r
En,q,ξ x ∞
Zp
···
Zp
q
n
n −1l qlx
l
l l
l0 1/1 q ξ r
n
nl −1l qlx
−qh−1l ξl ; q−1 r
2.8
l0
mr−1
m
q
qh−rm −1m x mnq,ξm .
h,r
m0
n
1−q
∞ r
Let Fq,ξ t, x n
En,q,ξ x1 tn /n!. By 2.7, we easily see that
r
j1 h−jxj
ξ
r
j1
xj
x x1 · · · xr nq dμ−1 x1 · · · dμ−1 xr .
2.9
Thus, we obtain the following theorem.
Theorem 2.2. For h ∈ Z, r ∈ N, n ≥ 0, and ε ∈ Tp , let
2r
∞ mr−1
m0
m
qh−rm −1m emxq,ξm t q
∞
n0
h,r
En,q,ξ x
tn
.
n!
2.10
Journal of Inequalities and Applications
7
Then
2r
h,r
En,q,ξ x 1−q
n
n
n −1l qlx
l h−rl ;q r
l0 −qh
2.11
m mr −1
r
h−rm
2
q
−1
−1m m xnq,ξm .
m
q
m0
∞
Now, we define the Nörlund-type twisted q-Euler polynomials of higher order as
follows:
h,−r
1
En,q,ξ x 1−q
h,−r
Let Fq,ξ
t, x ∞
n
n
l0 Zp
···
Zp
qlx1 ···xr q
nl −1l qlx
r
j1 h−jxj
ξlx1 ···xr dμ−1 x1 · · · dμ−1 xr .
2.12
h,−r
m0
En,q,ξ x1 tn /n!. By 2.12, we have
h,−r
Fq,ξ
t, x r
1
r
m
2 qh−rm
q
emxq,ξm t .
m q
2r m0
2.13
Thus, we obtain the following theorem.
Theorem 2.3. For h ∈ Z, r ∈ N, n ≥ 0, and ε ∈ Tp , let
∞
r
1
tn
r
h,−r
m
2 qh−rm
q
emxq,ξm t En,q,ξ x .
r
m q
2 m0
n!
n0
2.14
Then
h,−r
En,q,ξ x
1
n
2r 1 − q
n n
l0
l
−1l qlx −qh−rl ξl ; q
r
m
1
r
r
q 2 qh−rm
m xnq,ξm .
m
2 m0
q
r
2.15
8
Journal of Inequalities and Applications
For h r, we have
2r
r,r
En,q,ξ x 1−q
2
r
∞
n
n
nl −1l qlx
−qhl ; q r
l0
−1
m
m0
mr−1
m
2.16
m
−1 m q
xnq,ξm ,
n 1
n
−1l qlx −ql ξl ; q
n
r
2r 1 − q l0 l
r
m
1
r
r
q 2 qh−rm
m xnq,ξm .
m q
2 m0
r,−r
En,q,ξ x 2.17
Thus, it is easy to see that
qmx 2r
−qm−rl ; q r
Zp
Zp
···
Zp
···
q
r
j1 m−jxj mx
ξ
r
j1
xj
dμ−1 x1 · · · dμ−1 xr Zp
m r
r
x x1 · · · xr q q − 1 1 q j1 xj ξ j1 xj dμ−1 x1 · · · dμ−1 xr m r
r
l
m ···
q−1
x x1 · · · xr lq q j1 xj ξ j1 xj dμ−1 x1 · · · dμ−1 xr l
Zp
Zp
l0
m l 0,r
m q − 1 El,q,ξ x.
l
l0
2.18
Equation 2.18 implies that
m l 0,r
qmx 2r
m q − 1 El,q,ξ x.
m−r l
−q ξ; q r
l0
2.19
From 1.16, we derive
qh−1
Zp
···
−
Zp
Zp
r
j1 h−jxj
···
2
x 1 x1 · · · xr m
q q
Zp
x 1 x1 · · · xr m
q q
r
Zp
Zp
r
j1
j1 h−jxj
···
ξ
x 1 x1 · · · xr m
q q
r−1
xj
ξ
dμ−1 x1 · · · dμ−1 xr r
j1
xj
j1 h−1−jxj1
ξ
dμ−1 x1 · · · dμ−1 xr r−1
j1
xj1
dμ−1 x1 · · · dμ−1 xr .
2.20
Journal of Inequalities and Applications
9
By 2.20, we have
h,r
h,r
h−1,r−1
qh−1 En,q,ξ x 1 En,q,ξ x 2En,q,ξ
x.
2.21
By simple calculation, we see that
q
x
Zp
···
Zp
x1 · · · xr xnq q
q−1
Zp
j1 h−j1xj
ξ
r
j1
Zp
···
···
r
Zp
x1 · · · xr xn1
q q
Zp
x1 · · · xr xm
q q
xj
dμ−1 x1 · · · dμ−1 xr r
r
j1 h−jxj
j1 h−jxj
ξ
r
j1
xj
ξ
r
j1
xj
dμ−1 x1 · · · dμ−1 xr 2.22
dμ−1 x1 · · · dμ−1 xr .
By 2.22, we see that
h,r
h1,r
h,r
qx En,q,ξ x q − 1 En1,q,ξ x En,q,ξ x.
2.23
Therefore, we obtain the following theorem.
Theorem 2.4. For h ∈ Z, r ∈ N, n ≥ 0, and ε ∈ Tp ,
h,r
h,r
h−1,r−1
qh−1 En,q,ξ x 1 En,q,ξ x 2En,q,ξ
q
x
h1,r
En,q,ξ x
x ,
h,r
h,r
q − 1 En1,q,ξ x En,q,ξ x.
2.24
Moreover,
m l 0,r
qmx 2r
m q − 1 El,q,ξ x.
m−r
l
−q ξ; q r
l0
2.25
From 2.16, we note that
r,r
2r
En,q−1 ,ξ r − x 1−q
−1 n
r
n
nl −1l qlr−x
−l −1 −q ξ; q r
l0
2r
−1n qn 2 1−q
r
n
n
nl −1l qlx
l −q ξ; q r
l0
2.26
r,r
−1n qn 2 En,q,ξ x.
In the case x r, we obtain
r,r
r
r,r
En,q−1 ,ξ 0 −1n qn 2 En,q,ξ r.
2.27
10
Journal of Inequalities and Applications
From 2.21 with h r, we derive
r,r
r,r
r−1,r−1
qr−1 En,q,ξ x 1 En,q,ξ x 2En,q,ξ
x.
2.28
3. Further Remarks on Nörlund-Type Twisted q-Euler Polynomials
0,r
0,−r
In the case h 0, let us consider the polynomials En,q,ξ x and En,q,ξ x as follows:
0,r
En,q,ξ x 0,−r
En,q,ξ x
Zp
···
n
Zp
l0 Zp
q−
···
r
jxj
j1
ξ
r
j1
xj
x x1 · · · xr nq dμ−1 x1 · · · dμ−1 xr ,
n
Zp
qlx1 ···xr q−
r
j1
3.1
−1l qlx
l
jxj lx1 ···xr ξ
dμ
−1 x1 · · · dμ−1 xr .
Then we have
2r
0,r
En,q,ξ x n
1−q
∞ 2r
m0
0,−r
En,q,ξ x
1
n
nl −1l qlx
l−r l0 −q ξ; q r
mr−1
q−rm −1m x mnq,ξm ,
m
q
n
2r 1 − q
n n
l
l0
l lx
−1 q
−q
l−r l
ξ ;q
3.2
r
r
1
r
m
2 qh−rm
q
m xnq,ξm .
m q
2r m0
Let us consider the following polynomials:
h,1
En,q,ξ x Zp
2
qx1 h−1 ξx1 x x1 nq dμ−1 x1 1−q
n
n
nl −1l qlx
l0
1 qlh−1 ξl
.
3.3
By the simple calculation of fermionic p-adic invariant integral, we see that
qx
Zp
qx1 h−1 ξx1 x x1 nq dμ−1 x1 q − 1
Zp
q
Zp
qx1 h−2 ξx1 x x1 n1
q dμ−1 x1 3.4
x1 h−2 x1
ξ x x1 nq dμ−1 x1 .
Thus
h−1,1
h,1
h−1,1
qx En,q,ξ x q − 1 En1,q,ξ x En,q,ξ x.
3.5
Journal of Inequalities and Applications
11
It is easy to see that
Zp
qh−1x1 ξx1 x x1 nq dμ−1 x1 n n
j1
j
n−j
xq qjx
j
Zp
x1 q qh−1x1 ξx1 dμ−1 x1 .
3.6
By 2.20, we can obtain that
h,1
En,q,ξ x
n n
j
j0
n
n−j
h,1
h,1
xq qjx Ej,q,ξ qx Eq,ξ xq ,
n ≥ 0,
h,1
3.7
h,1
where we use the technique method notation by replacing Eq,ξ n by En,q,ξ , symbolically.
From 1.14, we can also derive
Zp
q
h−1x1 1 x1 1
ξ
x x1 nq qh−1x1 ξx1 dμ−1 x1 2xnq .
3.8
qh−1 En,q,ξ x 1 En,q,ξ x 2xnq .
h,1
3.9
n
h,1
h,1
qh−1 En,q,ξ 1 En,q,ξ 2δn,0 ,
3.10
x x1 1nq dμ−1 x1 Zp
Thus, we obtain that
h,1
For x 0, we have
where δn,0 is the Kronecker symbol. It is easy to see that
h,1
E0,q,ξ
Zp
qx1 h−1 ξx1 dμ−1 x1 2
2
.
h−1
1 q ξ 2qh−1 ,ξ
3.11
By 3.3, we see that
h,1
En,q,ξ 1 − x Zp
1 − x x1 nq−1 q−x1 h−1 ξx1 dμ−1 x1 2
−1n qnh−1 1−q
n
n
nl −1l qlx
l0
1
3.12
qlh−1 ξl
h,1
−1n qnh−1 En,q,ξ x.
In particular, for x 1, we obtain that
h,1
h,1
h,1
En,q,ξ 0 −1n qnh−1 En,q,ξ 1 −1n−1 qn En,q,ξ .
3.13
12
Journal of Inequalities and Applications
Acknowledgment
This paper was supported by Konkuk University in 2010.
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