Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 354329, 12 pages
doi:10.1155/2011/354329
Research Article
On the Higher-Order q-Euler Numbers and
Polynomials with Weight α
K.-W. Hwang,1, 2 D. V. Dolgy,3 T. Kim,2 and S. H. Lee2
1
Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 9 May 2011; Accepted 19 June 2011
Academic Editor: Cengiz Çinar
Copyright q 2011 K.-W. Hwang 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.
The main purpose of this paper is to present a systemic study of some families of higher-order
q-Euler numbers and polynomials with weight α. In particular, by using the fermionic p-adic qintegral on Zp , we give a new concept of q-Euler numbers and polynomials with weight α.
1. Introduction
Let p be a fixed odd prime. Throughout this paper Zp , Qp , C, and Cp , will, respectively, denote
the ring of p-adic rational integers, the field, of p-adic rational numbers, the complex number
field and the completion of algebraic closure of Qp . 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 p−1
see 1–14. When one speaks of q-extension, q can be regarded as an indeterminate, complex
number q ∈ C, or p-adic number q ∈ Cp ; it is always clear from context. If q ∈ C, we assume
|q| < 1. If q ∈ Cp , then we assume |1 − q|p < 1 see 1–14.
In this paper, we use the notation of q-number as follows:
xq 1 − qx
1−q
see 1–14. Note that limq → 1 xq x for any x with |x|p ≤ 1 in the p-adic case.
1.1
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Let CZp be the space of continuous functions on Zp . For f ∈ CZp , the fermionic
p-adic q-integral on Zp is defined by
I−q f −1
p
x
1
fxdμ−q x lim N fx −q ,
N
→
∞
p −q x0
Zp
N
1.2
lim
2q
N
p
−1
2
N →∞
x
fx −q
x0
see 4–7.
From 1.2, we note that
qI−q f1 I−q f 2q f0,
1.3
where f1 x fx 1.
It is well known that the ordinary Euler polynomials are defined by
∞
tn
2
xt
Ext
,
e
e
E
x
n
n!
et 1
n0
1.4
with the usual convention of replacing En x by En x.
In the special case, x 0 and En 0 En are called the nth Euler numbers see 1–14.
By 1.5, we get the following recurrence relation as follows:
E 1n E E0 1,
⎧
⎨2,
if n 0,
⎩0,
if n > 0.
1.5
Recently, h, q-Euler numbers are defined by
h
E0,q 2
,
1 qh
qh
⎧
⎨2,
n
h
h
qEq 1 Eq ⎩0,
h n
if n 0,
if n > 0,
1.6
h
with the usual convention about replacing Eq by En,q see 1–12.
h
Note that limq → 1 En,q En .
For α ∈ N, the weight q-Euler numbers are also defined by
α
E0,q 1,
⎧
⎨2q ,
n
α
α
q qα Eq 1 En,q ⎩0,
n
if n 0,
if n > 0,
α
α
with the usual convention about replacing Eq by En,q see 4.
1.7
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3
The purpose of this paper is to present a systemic study of some families of higherorder q-Euler numbers and polynomials with weight α. In particular, by using the fermionic
p-adic q-integral on Zp , we give a new concept of q-Euler numbers and polynomials with
weight α.
2. Higher-Order q-Euler Numbers and Polynomials with Weight α
For h ∈ Z, α, k ∈ N, and n ∈ Z , let us consider the expansion of higher-order q-Euler
polynomials with weight α as follows:
α
En,q h, k | x
···
x1 · · · xk xnqα qx1 h−1···xk h−k dμ−q x1 · · · dμ−q xk .
Zp
Zp
2.1
k-times
From 1.2 and 2.1, we note that:
α
En,q h, k
2kq
| x n
1 − qα
n
n
l0
α
l
−1l 1q
αlh
qαlx
.
· · · 1 qαlh−k1
2.2
α
In the special case, x 0, En,q h, k | 0 En,q h, k are called the higher-order q-Euler numbers
with weight α.
By 2.1, we get
α
α
α
En,q h, k qα − 1 En1,q h − α, k En,q h − α, k.
2.3
From 2.1 and 2.2, we have
α
E0,q mα, k 1
···
Zp
m
m l0
Zp
q
k1
j1 mα−jxj
l
l
qα − 1
dμ−q x1 · · · dμ−q xk1 Zp
···
Zp
x1 · · · xk1 lqα q−
m
m l0
l
1q
l α
qα − 1 El,q 0, k 1
2k1
q
.
1 qαm−1 · · · 1 qαm−k
αm
k1
j1
jxj
dμ−q x1 · · · dμ−q xk1 2.4
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Discrete Dynamics in Nature and Society
From 2.1, we can derive the following equation:
i
i j0
j
qα − 1
j
Zp
Zp
···
i−1
···
n−ij h−α−1x1 ···h−α−kxk
x1 · · · xk qα
Zp
q
dμ−q x1 · · · dμ−q xk h−1x1 ···h−kxk αx1 ···xk i−1
q
dμ−q x1 · · · dμ−q xk x1 · · · xk n−i
qα q
Zp
j
q −1
i−1
α
α
En−ij,q h, k.
j
j0
2.5
By 2.1, 2.2, 2.3, and 2.4, we see that
i
j
q −1
α
j0
i−1
i
j i − 1
α
α
α
q −1
En−ij,q h − α, k En−ij,q h, k.
j
j
j0
2.6
Therefore, we obtain the following theorem.
Theorem 2.1. For α, k ∈ N and n, i ∈ Z , one has
i
i j0
i−1
j α
j
α
q −1
q − 1 En−ij,q h − α, k j
α
i−1
α
En−ij,q h, k.
2.7
.
· · · 1 qαm−k1
2.8
j
j0
By simple calculation, we easily see that
m
m j0
j
j α
q − 1 Ej,q 0, k α
1 qαm
1q
2kq
αm−1
α
3. Polynomials En,q 0, k | x
α
We now consider the polynomials En,q 0, k | x in qx by
α
En,q 0, k | x Zp
···
Zp
x x1 · · · xk nqα q−
k
j1
jxj
dμ−q x1 · · · dμ−q xk .
3.1
k-times
By 3.1, we get
n
n αlx
n α
1
k
q −1n−l q − 1 En,q 0, k | x 2q
.
αl
1 q · · · 1 qαl−k1
l
l0
α
3.2
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5
From 3.1 and 3.2, we can derive the following equation:
Zp
···
Zp
q
k
j1 αn−jxj αnx
j0
dμ−q x1 · · · dμ−q xk n
n
j
j α
j
αq q − 1 Ej,q 0, k | x,
3.3
2kq qαnx
···
q j1 αn−jxj αnx dμ−q x1 · · · dμ−q xk .
1 qαn · · · 1 qαn−k1
Zp
Zp
k
Therefore, by 3.2 and 3.3, we obtain the following theorem.
Theorem 3.1. For α ∈ N and n, k ∈ Z , one has
α
En,q 0, k | x n
n
l0
l
n
2kq
n
1
−1l qαlx αl−k1 ,
n
n
:q k
−q
αq 1 − q l0 l
αnx
3.4
2kq
q
l α
αlq q − 1 El,q 0, k | x αn−k1 ,
:q k
−q
where a : q0 1 and a : qk 1 − a1 − aq · · · 1 − aqk−1 .
Let d ∈ N with d ≡ 1 mod2. Then we have
⎡
Zp
···
Zp
⎣x k
⎤n
xj ⎦ q−
j1
dnqα
d−1
k
j1
×
Zp
···
dμ−q x1 · · · dμ−q xk qα
q−
k
j2 j−1aj
dk−q a1 ,...,ak 0
jxj
⎡
⎣
x
−1
k
j1 aj
d
Zp
k
j1
aj
3.5
⎤n
k
k
xj ⎦ q−d j1 jxj dμ−qd x1 · · · dμ−qd xk .
j1
qαd
Thus, by 3.5, we obtain the following theorem.
Theorem 3.2. For d ∈ N with d ≡ 1 mod 2, one has
α
En,q 0, k | x dnqα
d−1
q−
dk−q a1 ,...,ak 0
k
j2 j−1aj
−1
k
j1
aj
x a1 · · · ak α
.
En,qd 0, k |
d
3.6
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Moreover,
α
En,q 0, k | dx dnqα
d−1
q−
k
j2 j−1aj
dk−q a1 ,...,ak 0
−1
k
j1
aj
a1 · · · ak α
.
En,qd 0, k | x d
3.7
By 3.1, we get
α
En,q 0, k
| x n
n
l
l0
α
α
αlx El,q 0, k,
xn−l
qα q
3.8
α
where En,q 0, k | 0 En,q 0, k.
Thus, we note that
α En,q 0, k
n
n n−l αly α
|xy y qα q El,q 0, k | x.
l
l0
3.9
α
4. Polynomials En,q h, 1 | x
α
Let us define polynomials En,q h, 1 | x as follows:
α
En,q h, 1 | x Zp
x x1 nqα qx1 h−1 dμ−q x1 .
4.1
From 4.1, we have
2q
α
En,q h, 1 | x 1−
n
qα
n
n
l0
l
1
.
1 qαlh
−1l qαlx 4.2
By the calculation of the fermionic p-adic q-integral on Zp , we see that
qαx
Zp
x x1 nqα qx1 h−1 dμ−q x1 q −1
α
Zp
x x1 h−α−1
dμ−q x1 x1 n1
qα q
4.3
Zp
x x1 nqα qx1 h−α−1 dμ−q x1 .
Thus, by 4.3, we obtain the following theorem.
Theorem 4.1. For α ∈ N and h ∈ Z, one has
α
α
α
qαx En,q h, 1 | x qα − 1 En1,q h − α − 1, 1 | x En,q h − α − 1, 1 | x.
4.4
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It is easy to show that
α
En,q h, 1 | x Zp
n
n
l0
x x1 nqα qx1 h−1 dμ−q x1 αlx
xn−l
qα q
l
Zp
n
n
l0
x1 lqα qx1 h−1 dμ−q x1 4.5
α
αlx El,q h, 1
xn−l
qα q
l
n
α
qαx Eq h, 1 xqα ,
for n ≥ 1,
n
α
α
with the usual convention about replacing Eq h, 1 by En,q h, 1.
From qI−q f1 I−q f 2q f0, we have
qh
Zp
x x1 1nqα qx1 h−1 dμ−q x1 Zp
x x1 nqα qx1 h−1 dμ−q x1 2q xnqα .
4.6
By 4.3 and 4.6, we get
α
α
qh En,q h, 1 | x 1 En,q h, 1 | x 2q xnqα .
4.7
For x 0 in 4.7, we have
α
α
qh En,q h, 1 | 1 En,q h, 1 ⎧
⎨2q ,
if n 0,
⎩0,
if n > 0.
4.8
Therefore, by 4.8, we obtain the following theorem.
Theorem 4.2. For h ∈ Z and n ∈ Z , one has
qh
⎧
⎨2q ,
n
α
α
qα Eq h, 1 1 En,q h, 1 ⎩0,
α
n
if n 0,
if n > 0,
4.9
α
with the usual convention about replacing Eq h, 1 by En,q h, 1.
From the fermionic p-adic q-integral on Zp , we easily get
α
E0,q h, 1 Zp
qx1 h−1 dμ−q x1 2q
2qh
.
4.10
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Discrete Dynamics in Nature and Society
By 4.1, we see that
α
En,q−1 h, 1 | 1 − x Zp
1 − x x1 nq−α q−x1 h−1 dμ−q−1 x1 n αnh−1
−1 q
2q
1−
n
qα
n
n
l0
l
−1l qαlx
1
1 qαlh
4.11
α
−1n qαnh−1 En,q−1 h, 1 | x.
Therefore, by 4.11, we obtain the following theorem.
Theorem 4.3. For α ∈ N, h ∈ Z, and n ∈ Z , one has
α
α
En,q−1 h, 1 | 1 − x −1n qαnh−1 En,q h, 1 | x.
4.12
In particular, for x 1, one gets
α
α
En,q h, 1 −1n qαnh−1 En,q h, 1 | 1
−1
n−1 αn−1
q
α
En,q h, 1
4.13
if n ≥ 1.
Let d ∈ N with d ≡ 1 mod 2. Then one has
Zp
qx1 h−1 x x1 nqα dμ−q x1 d−1
n
dnqα xa
a
ha
x1
q −1
qx1 h−1d dμ−qd x1 .
d
d−q a0
qαd
Zp
4.14
Therefore, by 4.14, we obtain the following theorem.
Theorem 4.4 Multiplication formula. For d ∈ N with d ≡ 1 mod 2, we have
α
En,q h, 1 | x d−1
dnqα x a
α
.
qha −1a En,qd h, 1 |
d
d−q a0
4.15
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9
5. Polynomials En,q h, k | x and k h
α
In 2.1, we know that
α
En,q h, k | x
···
Zp
Zp
x1 · · · xk xnqα qh−1x1 ···h−kxk dμ−q x1 · · · dμ−q xk .
5.1
Thus, we get
n
n
n α
α
qαlx
k
q − 1 En,q h, k | x 2q
,
−1n−l 1 qαlh · · · 1 qαlh−k1
l
l0
q
h
Zp
···
Zp
x 1 x1 · · · xk nqα qh−1x1 ···h−kxk dμ−q x1 · · · dμ−q xk −
5.2
···
Zp
Zp
x x1 · · · xk nqα qh−1x1 ···h−kxk dμ−q x1 · · · dμ−q xk 2q
Zp
···
Zp
x x2 · · · xk nqα qh−2x2 ···h−kxk dμ−q x2 · · · dμ−q xk .
Therefore, by 2.1 and 5.2, we obtain the following theorem.
Theorem 5.1. For h ∈ Z, α ∈ N, and n ∈ Z , one has
α
α
α
n,q h − 1, k − 1 | x.
qh En,q h, k | x 1 En,q h, k | x 2q E
5.3
Note that
q
αx
Zp
···
Zp
x x1 · · · xk nqα qhx1 h−1x2 ···h1−kxk dμ−q x1 · · · dμ−q xk q −1
Zp
α
Zp
···
Zp
···
Zp
h−αx1 ···h1−α−kxk
dμ−q x1 · · · dμ−q xk x x1 · · · xk n1
qα q
x x1 · · · xk nqα qh−αx1 ···h1−α−kxk dμ−q x1 · · · dμ−q xk α
α
qα − 1 En1,q h 1 − α, k | x En,q h 1 − α, k | x.
Therefore, by 5.4, we obtain the following theorem.
5.4
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Theorem 5.2. For n ∈ Z , one has
α
α
α
qαx En,q h 1, k | x qα − 1 En1,q h 1 − α, k | x En,q h 1 − α, k | x.
5.5
Let d ∈ N with d ≡ 1 mod 2. Then we get
⎡
···
Zp
Zp
⎤n
k
k
⎣x xj ⎦ q j1 h−jxj dμ−q x1 · · · dμ−q xk j1
dnqα
qα
d−1
qh
k
j1
aj −
k
j2 j−1aj
dk−q a1 ,...,ak 0
×
⎡
Zp
···
⎣
x
k
j1
aj
d
Zp
−1
k
k
j1
aj
⎤n
xj ⎦
j1
qd
5.6
k
j1 h−jxj
dμ−qd x1 · · · dμ−qd xk .
qαd
Therefore, by 5.6, we obtain the following theorem.
Theorem 5.3. For d ∈ N with d ≡ 1 mod 2, one has
α
En,q h, k | dx
dnqα
d−1
qh
k
j1
aj −
k
j2 j−1aj
dk−q a1 ,...,ak 0
α
−1
k
j1
aj
a1 · · · ak α
.
En,qd h, k | x d
5.7
α
Let En,q k, k | x En,q k | x. Then we get
n α
qα − 1 En,q k | x,
n
n
l0
Zp
···
Zp
l
2kq
1 qαlk · · · 1 qαl1
−1n−l qαlx k − x x1 · · · xk nq−α q−k−1x1 −···−k−kxk dμ−q−1 x1 · · · dμ−q−1 xk α k1 −k
n
n
q 2
1
k
−1l qαlx n 2q
αl1
−α
1q
· · · 1 qαlk
1−q
l
l0
−1n qnα q
−1n q
α k1 −k
2
2kq
1−q
α n
α n k1 −k α
2
En,q k
n
l0
| x.
nl −1l qαlx
1 qαl1 · · · 1 qαlk
5.8
Discrete Dynamics in Nature and Society
11
Therefore, by 5.8, we obtain the following theorem.
Theorem 5.4. For n ∈ Z , one has
α
α
En,q−1 k | k − x −1n q
n k1 −k α
2
En,q k
| x.
5.9
Let x k in Theorem 5.4. Then we see that
n k1 −k α
2
En,q k
α
α
En,q−1 k | 0 −1n q
| k.
5.10
From 4.6 and Theorem 5.1, we note that
α
α
α
qk En,q k | x 1 En,q k | x 2q En,q k − 1 | x.
5.11
It is easy to show that
n
2kq
n
n α
ln
q − 1 En,q k | 0 .
−1 1 qαl1 · · · 1 qαlk
l
l0
α
5.12
By simple calculation, we get
n
n l0
l
l
qα − 1
···
Zp
Zp
x1 · · · xk lqα q
k
l1 k−lxl
dμ−q x1 · · · dμ−q xk 5.13
1 qαnk
2kq
.
1 qαnk−1 · · · 1 qαn1
From 5.13, we note that
n
n l0
l
l α
qα − 1 El,q k | 0 1q
α
En,q k | x Zp
···
Zp
2kq
αnk
,
1 qαnk−1 · · · 1 qαn1
x x1 · · · xk nqα qk−1x1 ···k−kxk dμ−q x1 5.14
· · · dμ−q xk n
n
l0
l
α
α
qαlx El,q k | 0xn−l
qα
qxα Eq k | 0 xqα
n
for n ∈ Z ,
n
α
α
with the usual convention about replacing Eq k | 0 by En,q k | 0.
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Discrete Dynamics in Nature and Society
Put x 0 in 5.11; we get
α
α
α
qk En,q k | 1 En,q k | 0 2q En,q k − 1 | 0.
5.15
n
α
α
α
qk qα Eq k | 0 1 En,q k | 0 2q En,q k − 1 | 0,
5.16
Thus, we have
n
α
α
with the usual convention about replacing Eq k | 0 by En,q k | 0.
Acknowledgment
This work was supported by the Dong-A University research fund.
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Advances in
Operations Research
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