Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 136532, 8 pages
doi:10.1155/2009/136532
Research Article
Note on the q-Extension of Barnes’ Type Multiple
Euler Polynomials
Leechae Jang,1 Taekyun Kim,2 Young-Hee Kim,2 and
Kyung-Won Hwang3
1
Department of Mathematics and Computer Science, Konkuk University, Chungju 130-701, South Korea
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
3
Department of General Education, Kookmin University, Seoul 136-702, South Korea
2
Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr and
Kyung-Won Hwang, khwang7@kookmin.ac.kr
Received 30 August 2009; Accepted 28 September 2009
Recommended by Vijay Gupta
We construct the q-Euler numbers and polynomials of higher order, which are related to Barnes’
type multiple Euler polynomials. We also derive many properties and formulae for our q-Euler
polynomials of higher order by using the multiple integral equations on Zp .
Copyright q 2009 Leechae Jang 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.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, symbols Z, Zp , Qp , and Cp will
denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational
numbers, and the completion of algebraic closure of Qp , respectively. Let N be the set of
natural numbers and Z N ∪ {0}. Let νp be the normalized exponential valuation of Cp
with |p|p p−νp 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 , then one normally assumes |1 − q|p < 1. We use the following
notations:
xq for all x ∈ Zp see 1–6.
1 − qx
,
1−q
x−q 1 − −qx
,
1q
1.1
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Journal of Inequalities and Applications
Let d a fixed positive odd integer with p, d 1. For N ∈ N, we set
X Xd →Z
lim−N
,
X1 Zp ,
dpN Z
a dp Zp ,
X∗ 1.2
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a moddpN ,
where a ∈ Z lies in 0 ≤ a < dpN . The fermionic p-adic q-measures on Zp are defined as
a
−q
μ−q a dpN Zp N ,
dp −q
1.3
see 5.
We say that f is a uniformly differentiable function at a point a ∈ Zp and write f ∈
UDZp , if the difference quotients Ff x, y fx − fy/x − y have a limit f a as
x, y → a, a. For f ∈ UDZp , let us begin with expression
p
1
N
−qj f j f j μ−q j pN Zp ,
−q 0≤j<pN
1.4
0≤j<pN
which represents a q-analogue of Riemann sums for f in the fermionic sense see 4, 5. The
integral of f on Zp is defined by the limit of these sums as n → ∞ if this limit exists. The
fermionic invariant p-adic q-integral of function f ∈ UDZp is defined as
I−q f Zp
fxdμ−q x lim
N
p
−1
2q
N →∞1
qp
fx−qx .
N
1.5
x0
Note that if fn → f in UDZp , then
Zp
fn xdμ−q x −→
Zp
fxdμ−q x fxdμ−q x,
X
Zp
fxdμ−q x.
1.6
The Barnes’ type Euler polynomials are considered as follows:
k
2
k j1
∞
tn
k
xt
,
E
,
.
.
.
,
w
|
w
e
x
1
k
n
n!
e wj t 1
n0
1
where w1 , w2 , . . . , wk ∈ Z cf. 7.
1.7
Journal of Inequalities and Applications
3
From 1.5, we can derive the fermionic invariant integral on Zp as follows:
lim I−q f I−1 f q→1
Zp
1.8
fxdμ−1 x.
For n ∈ N, let fn x fx n, one has
n−1
I−1 fn −1n I−1 f 2 −1n−1−l fl.
1.9
l0
By 1.9, we see that
e
xt
Zp
···
Zp
e
w1 x1 ···wk xk t
dμ−1 x1 · · · dμ−1 xk 2
k
k j1
1
e wj t 1
.
1.10
k-times
From 1.10, we note that
Zp
···
k
Zp
x w1 x1 · · · wk xk n dμ−1 x1 · · · dμ−1 xk En x | w1 , . . . , wk .
1.11
k-times
In the view point of 1.11, we try to study the q-extension of Barnes’ type Euler polynomials
by using the q-extension of fermionic p-adic invariant integral on Zp .
The purpose of this paper is to construct the q-Euler numbers and polynomials of
higher order, which are related to Barnes’ type multiple Euler numbers and polynomials.
Also, we give many properties and formulae for our q-Euler polynomials of higher order.
Finally, we give the generating function for these q-Euler polynomials of higher order.
2. Barnes’ Type Multiple q-Euler Polynomials
Let a1 , a2 , . . . , ak , b1 , b2 , . . . , bk ∈ Z. For w ∈ Zp and q ∈ Cp with |1 − q|p < 1, we define the
Barnes’ type multiple q-Euler polynomials as follows:
k
En,q w
| a1 , . . . , ak ; b1 , . . . , bk ⎤n
k
q j1 bj −1xj ⎣w aj xj ⎦ dμ−q x,
k
Zkp
⎡
j1
q
2.1
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Journal of Inequalities and Applications
where
Zkp
fx1 , . . . , xk dμ−q x Zp
···
Zp
fx1 , . . . , xk dμ−q x1 · · · dμ−q xk 2.2
k-times
see 1, 5.
k
In the special case w 0, En a1 , . . . , ak ; b1 , . . . , bk are called the Barnes’ type multiple
q-Euler numbers. From 2.1, one has
Zkp
q
⎡
k
j1 bj −1xj
⎣w k
⎤n
aj xj ⎦ dμ−q x
j1
q
n
n 1
1 − qn r0
−q
r
n
n 1
1 − qn r0
−q
r
w r
lim
N →∞
w r
2kq
k
1q
1
N
qp
N
p
−1
k
j1 aj rbj xj
2.3
−1x1 ···xk
x1 ,...,xk 0
1
1 qaj rbj
j1
q
.
Therefore, we obtain the following theorem.
Theorem 2.1. Let w ∈ Zp and k ∈ N. For a1 , a2 , . . . , ak , b1 , b2 , . . . , bk ∈ Z, one has
k
n
2kq n w r 1
.
| a1 , . . . , ak ; b1 , . . . , bk −q
n
1 qaj rbj
1 − q r0 r
j1
k
En,q w
2.4
By 1.7, we easily see that
k
k
lim En,q w | a1 , . . . , ak ; b1 , . . . , bk En w | a1 , . . . , ak .
2.5
q→1
From 1.7, we can derive
Zkp
q
⎡
⎤n
k
aj xj ⎦ dμ−q x
k
j1 bj −1xj ⎣
j1
q−1
Zkp
q
q
k
⎡
k
j1 bj −aj −1xj ⎣
j1
⎤n1
aj xj ⎦
dμ−q x q
Zkp
q
k
⎡
⎤n
k
aj xj ⎦ dμ−q x.
j1 bj −aj −1xj ⎣
j1
q
2.6
Journal of Inequalities and Applications
5
By 2.6, one has
k
k
En,q a1 , . . . , ak ; b1 , . . . , bk q − 1 En1,q a1 , . . . , ak ; b1 − a1 , . . . , bk − ak k
2.7
En,q a1 , . . . , ak ; b1 − a1 , . . . , bk − ak .
Hence we obtain the following theorem.
Theorem 2.2. For k ∈ N and n ∈ Z , one has
k
k
En,q a1 , . . . , ak ; b1 , . . . , bk q − 1 En1,q a1 , . . . , ak ; b1 − a1 , . . . , bk − ak k
2.8
En,q a1 , . . . , ak ; b1 − a1 , . . . , bk − ak .
It is not difficult to show that the following integral equation is satisfied:
i
i
j0
q − 1j
j
n−ij k bl −1xl
l1
Zkp
a1 x1 · · · ak xk q
i
i−m
dμ−q x
2.9
q − 1j
j
j0
q
Zkp
a1 x1 · · · n−ij k
ak xk q q l1 bl mal −1xl dμ−q x,
where m ∈ N with i ≥ m. By 2.9, we obtain the following theorem.
Theorem 2.3. Let n ∈ N and m ∈ N. For i ∈ N with i ≥ m, one has
i
i
j0
k
q − 1j En−ij,q a1 , . . . , ak ; b1 , . . . , bk j
2.10
i
i−m
j0
k
q − 1j En−ij,q a1 , . . . , ak ; b1 ma1 , . . . , bk mak .
j
For the special case k 1 in Theorem 2.3, one has
n
n
j0
j
q − 1
Zp
j
1
Ej,q a1 ; b1 n
n−m
j0
qna1 b1 −1x dμ−q x j
2q
1 qna1 b1
1
q − 1j Ej,q a1 ; b1 ma1 2.11
.
By 2.1, 2.3, and 2.9, we obtain the following a corollary.
6
Journal of Inequalities and Applications
Corollary 2.4. For n, k ∈ N and w ∈ Zp , one has
k
En,q w
k
n
2kq n w r 1
| a1 , . . . , ak ; b1 , . . . , bk −q
n
1 qaj rbj
1 − q r0 r
j1
n
n
k
wi
wn−i
q q Ei,q w | a1 , . . . , ak ; b1 , . . . , bk .
i
i0
2.12
From 2.3, we note that
⎡
qw
Zkp
⎤m
k
k
⎣w aj xj ⎦ q j1 bj −1xj dμ−q x
j1
q−1
⎡
⎣w Xk
Zkp
⎣w Zkp
k
k
⎤m1
aj xj ⎦
j1
⎡
q
⎡
⎣w k
⎤m
aj xj ⎦ q
j1
j1 bj −aj −1xj
dμ−q x
aj xj ⎦ q
k
j1 bj −aj −1xj
dμ−q x,
q
2.13
k
j1 bj −1xj
dμ−q x
q
k
dm
q 2q
× −1
k
q
⎤m
j1
q
d−1
q
k
j1
bj ij
i1 ,...,ik 0
i1 ···ik
⎡
⎣
w
Zkp
k
j1
d
aj ij
k
⎤m
aj xj ⎦
j1
qd
k
j1 bj −1xj
dμ−qd x,
qd
where d is an odd positive integer. By 2.13, we obtain the following theorem.
Theorem 2.5. For d ∈ N with d ≡ 1 (mod2), one has
k
Em,q w | a1 , . . . , ak ; b1 , . . . , bk k
dm
q 2q
d−1
q
k
j1 bj ij
⎛
−1
i1 ,...,ik 0
i1 ···ik
k
Em,qd ⎝
w
k
j1
d
k
qw Em,q w | a1 , . . . , ak ; b1 , . . . , bk k
q − 1 Em1,q w | a1 , . . . , ak ; b1 − a1 , . . . , bk − ak k
Em,q w | a1 , . . . , ak ; b1 − a1 , . . . , bk − ak .
aj ij
⎞
| a1 , . . . , ak ; b1 , . . . , bk ⎠,
2.14
Journal of Inequalities and Applications
7
Remark 2.6. Let
Fq w, t ∞
tn
k
En,q w | a1 , . . . , ak ; b1 , . . . , bk .
n!
n0
2.15
From 2.4, we can easily derive the following equation:
∞
tm
k
Fq w, q − 1 t q − 1m Em,q w | a1 , . . . , ak ; b1 , . . . , bk m!
m0
⎛
⎞
k
∞
i
1
⎠qwi t .
2kq e−t ⎝
aj ibj
i!
i0
j1 1 q
2.16
By differentiating both sides of 2.16 with respect to t and comparing coefficients on both
sides, one has
k
k
qw Em,q w | a1 , . . . , ak ; b1 , . . . , bk − Em,q w | a1 , . . . , ak ; b1 − a1 , . . . , bk − ak k
q − 1 Em1,q w | a1 , . . . , ak ; b1 − a1 , . . . , bk − ak .
2.17
The inversion formula of Equation 2.4 at w 0 is given by
m
m
i0
i
q − 1i
Zkp
a1 x1 · · · ak xk iq q
k
j1 bj −1xj
dμ−q x Zkp
q
k
j1 maj bj −1xj
dμ−q x.
2.18
Thus, one has
m
m
i0
i
q − 1
i
k
Ei,q a1 , . . . , ak ; b1 , . . . , bk 2kq
k
j1
1
1 qmaj bj
.
2.19
Acknowledgment
This paper was supported by Konkuk University 2009.
References
1 M. Acikgoz and Y. Simsek, “On multiple interpolation functions of the Nörlund-type q-Euler
polynomials,” Abstract and Applied Analysis, vol. 2009, Article ID 382574, 14 pages, 2009.
2 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-genocchi numbers,” Advanced
Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
3 N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer operators,” Advanced
Studies in Contemporary Mathematics, vol. 19, pp. 97–108, 2009.
4 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
8
Journal of Inequalities and Applications
5 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the
fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 17, 2010.
6 Y.-H. Kim, W. Kim, and C. S. Ryoo, “On the twisted q-Euler zeta function associated with twisted
q-Euler numbers,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 1, pp. 93–100, 2009.
7 Y. Simsek, T. Kim, and I. S. Pyung, “Barnes’ type multiple Changhee q-zeta functions,” Advanced Studies
in Contemporary Mathematics, vol. 10, no. 2, pp. 121–129, 2005.