Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 298531, 11 pages
doi:10.1155/2012/298531
Research Article
Explicit Formulas Involving q-Euler Numbers and
Polynomials
Serkan Araci,1 Mehmet Acikgoz,1 and Jong Jin Seo2
1
Department of Mathematics, Faculty of Science and Arts, University of Gaziantep,
27310 Gaziantep, Turkey
2
Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
Correspondence should be addressed to Jong Jin Seo, seo2011@pknu.ac.kr
Received 4 April 2012; Accepted 4 October 2012
Academic Editor: Gerd Teschke
Copyright q 2012 Serkan Araci 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.
We deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for qEuler numbers and polynomials by using their generating function and derivative operator. Also,
we derive relations between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral
in the p-adic integer ring.
1. Preliminaries
Imagine that p is a fixed odd prime number. Throughout this paper we use the following
notations, where Zp denotes the ring of p-adic rational integers, Q denotes the field of rational
numbers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of
algebraic closure of Qp . Let N be the set of natural numbers and N∗ N ∪ {0}.
The p-adic absolute value is defined by
p 1 .
p
p
1.1
In this paper, we will assume that |q − 1|p < 1 as an indeterminate.
xq is a q-extension of x, which is defined by
xq We note that limq → 1 xq x see 1–12.
1 − qx
.
1−q
1.2
2
Abstract and Applied Analysis
We say that f is a uniformly differentiable function at a point a ∈ Zp , if the difference
quotient
fx − f y
,
Ff x, y x−y
1.3
has a limit f a as x, y → a, a and denote this by f ∈ UDZp .
Let UDZp be the set of uniformly differentiable function on Zp . For f ∈ UDZp , let
us start with the expression
1
fξqξ fξμq ξ pN Zp ,
N
p q 0≤ξ<pN
0≤ξ<pN
1.4
which represents p-adic q-analogue of Riemann sums for f. The integral of f on Zp will be
defined as the limit N → ∞ of these sums, when it exists. The p-adic q-integral of function
f ∈ UDZp is defined by Kim
Iq f Zp
fξdμq ξ lim N →∞
N
−1
p
1
p
N
fξqξ .
1.5
q ξ0
The bosonic integral is considered as a bosonic limit q → 1, I1 f limq → 1 Iq f.
Similarly, the fermionic p-adic integral on Zp is introduced by Kim as follows:
I−q f lim Iq f q → −q
Zp
fξdμ−q ξ
for more details, see 9–12.
In 6, the q-Euler polynomials with weight 0 are introduced as
n
x y dμ−q y .
En,q x Zp
1.6
1.7
From 1.7, we have
En,q x n n
l0
l
xl En−l,q ,
1.8
where En,q 0 En,q are called q-Euler numbers with weight 0. Then, q-Euler numbers are
defined as
n
q Eq 1 En,q 2q ,
0,
if n 0,
if n /
0,
n
where the usual convention about replacing Eq by En,q is used.
1.9
Abstract and Applied Analysis
3
Similarly, the q-Bernoulli polynomials and numbers with weight 0 are defined,
respectively, as
p −1
n
1 x y qy
Bn,q x lim n n→∞ p
q y0
n
x y dμq y ,
n
Bn,q 1.10
Zp
Zp
yn dμq y
for more information, see 4.
We, by using the Kim et al. method in 2, will investigate some interesting identities
on the q-Euler numbers and polynomials arising from their generating function and
derivative operator. Consequently, we derive some properties on the q-Euler numbers and
polynomials and q-Bernoulli numbers and polynomials by using q-Volkenborn integral and
fermionic p-adic q-integral on Zp .
2. On the q-Euler Numbers and Polynomials
Let us consider Kim’s q-Euler polynomials as follows:
q
2q
q
Fx Fx t qet
1
ext ∞
tn
En,q x .
n!
n0
2.1
Here x is a fixed parameter. Thus, by expression of 2.1, we can readily see the
following:
q
q
qet Fx Fx 2q ext .
2.2
Last from equality, taking derivative operator D as D d/dt on the both sides of 2.2.
Then, we easily see that
q
q
qet D Ik Fx Dk Fx 2q xk ext ,
2.3
where k ∈ N∗ and I is identity operator. By multiplying e−t on both sides of 2.3, we get
q
q
qD Ik Fx e−t Dk Fx 2q xk ex−1t .
2.4
Let us take derivative operator Dm m ∈ N on both sides of 2.4. Then we get
q
q
qet Dm D Ik Fx Dk D − Im Fx 2q xk x − 1m ext .
2.5
4
Abstract and Applied Analysis
Let G0 not G0 be the constant term in a Laurent series of Gt. Then, from 2.5,
we get
k k
j0
j
t
qe D
km−j
q
Fx t
0 m m
j
j0
q
−1j Dkm−j Fx t 0 2q xk x − 1m .
2.6
By 2.1, we see
q
DN Fx t 0 EN,q x,
q
et DN Fx t 0 EN,q x.
2.7
By expressions of 2.6 and 2.7, we see that
max{k,m}
j0
q
k
m −1j
Ekm−j,q x 2q xk x − 1m .
j
j
2.8
From 2.1, we note that
n−1 d n − 1 n−1−l
nEn−1,q x.
En,q x n
El,q x
l
dx
l0
2.9
By 2.9, we easily see
1
2q−1
En1,q 1 − En1,q
−
En1,q .
n1
n1
En,q xdx 0
2.10
Now, let us consider definition of integral from 0 to 1 in 2.8, then we have
− 2q−1
max{k,m}
Ekm−j1,q
k
j m
q
−1
j
j
km−j 1
j0
2q −1m Bk 1, m 1
2q −1m
2.11
Γk 1Γm 1
,
Γk m 2
where Bm, n is beta function which is defined by
Bm, n 1
xm−1 1 − xn−1 dx
0
ΓmΓn
,
Γm n
2.12
m > 0, n > 0.
As a result, we obtain the following theorem.
Abstract and Applied Analysis
5
Theorem 2.1. For n ∈ N, one has
max{k,m}
q
j1
Ekm−j1,q
k
m
−1j
j
j
km−j 1
Ekm1,q
−1
.
q
km − 2q
km1
k m 1 k
2.13
m1
Substituting m k 1 into Theorem 2.1, we readily get
k1 E2k2−j,q
k
k1
q
−1j
j
j
2k 2 − j
j1
E2k2,q
−1
.
q
2k1 − 2q
2k 2
2k 2 k
2.14
k
By 2.1, it follows that
max{k,m}
km−j
j0
m k
Ekm−j−1,q x
q
−1j
j
j
2.15
2q xk−1 x − 1m−1 k mx − k.
Let m k in 2.1, we see that
k k
k q
−1j
E2k−j,q x 2q xk x − 1k .
j
j
j0
2.16
Last from equality, we discover the following:
2q
k/2
j0
k k/2
k E2k−2j,q x q − 1
E2k−2j−1,q x 2q xk x − 1k .
2j
2j
1
j0
2.17
Here · is Gauss’ symbol. Then, taking integral from 0 to 1 in both sides of last
equality, we get
k/2
k E2k−2j1,q
k E2k−2j,q
k/2
2q−1 1 − q
− 2q−1 2q
2j 2k − 2j 1
2j 1 2k − 2j
j0
j0
2q −1k Bk 1, k 1
2q −1k
.
2k 1 2k
k
Consequently, we derive the following theorem.
2.18
6
Abstract and Applied Analysis
Theorem 2.2. The following identity
2q
k/2
k E2k−2j,q
k/2
k E2k−2j1,q
q−1
2j 2k − 2j 1
2j 1 2k − 2j
j0
j0
2.19
q−1k1
2k 1 2k
k
is true.
In view of 2.1 and 2.17, we discover the following applications:
k1 k
k1 q
−1j
E2k1−j,q x
j
j
j0
2q E2k1,q x k1/2
q
j1
k1/2
k
k
k
−
−
E2k−2j,q x
2j 1
2j 1
2j
q
j0
⎡
k/2
−⎣
j0
2q
k
k
k
E2k1−2j,q x
2j
2j
2j − 1
⎤
k q − 1 k/2
k E
E2k−2j1 x⎦
x 2j 2k−2j,q
1 q j0 2j 1
k/2
j0
2.20
k/2
k k E2k1−2j,q x E2k1−2j,q x
2j
2j
−
1
j1
k/2
q−1
j0
k q − 1 k/2
k
E2k−2j,q x E2k−2j1 x.
2j 1
1 q j0 2j 1
By expressions 2.17 and 2.20, we have the following theorem.
Theorem 2.3. For k ∈ N, one has
2q
k/2
j0
k/2
k k E2k1−2j,q x E2k1−2j,q x
2j
2j
−
1
j1
k k/2
1 q−1
E2k−2j1 x
E2k−2j,q x 2j 1
1q
j0
xk x − 1k 2q x − q .
2.21
Abstract and Applied Analysis
7
3. p-adic Integral on Zp Associated with Kim’s q-Euler Polynomials
In this section, we consider Kim’s q-Euler polynomials by means of p-adic q-integral on Zp .
Now we start with the following assertion.
Let m, k ∈ N. Then by 2.8,
I1 2q
2q
Zp
m m
l0
2q
xk x − 1m dμ−q x
l
m m
l0
l
−1
m−l
Zp
xlk dμ−q x
3.1
−1m−l Elk,q .
On the other hand, in right hand side of 2.8,
I2 max{k,m}
j0
max{k,m}
j0
q
km−j
k m − j k
m
xl dμ−q x
−1j
Ekm−j−l,q
j
j
l
Zp
l0
km−j
k m − j k
j m
q
−1
Ekm−j−l,q El,q .
j
j
l
l0
3.2
Equating I1 and I2 , we get the following theorem.
Theorem 3.1. For m, k ∈ N, one has
max{k,m}
j0
km−j
k m − j k
j m
q
−1
Ekm−j−l,q El,q
j
j
l
l0
2q
m m
l0
l
3.3
−1m−l Elk,q .
Let us take fermionic p-adic q-integral on Zp in left hand side of 2.21, we get
I3 Zp
xk x − 1k 2q x − q dμ−q x
2q
k k k
k
xkl1 dμ−q x − q
xkl dμ−q x
−1k−l
−1k−l
l
l
Zp
Zp
l0
l0
2q
k k k
k
−1k−l Ekl1,q − q
−1k−l Ekl,q .
l
l
l0
l0
3.4
8
Abstract and Applied Analysis
In other words, we consider right hand side of 2.21 as follows:
I4 2q
k/2
j0
k/2
j1
k
2j
2k−2j1
l0
2k − 2j 1 xl dμ−q x
E2k1−2j−l,q
l
Zp
2k−2j1
2k − 2j 1
k
xl dμ−q x
E2k1−2j−l,q
2j − 1
l
Z
p
l0
⎤
2k − 2j 2k−2j
l
⎢
x dμ−q x ⎥
E2k−2j−l,q
q−1
k/2
⎥
k ⎢
l
Zp
⎥
⎢
j0
⎥
⎢
2k−2j1
⎥
⎢
2j
1
q
−
1
2k − 2j 1 j0
⎣
l
x dμ−q x⎦
E2k−2j−l1
l
1 q l0
Zp
⎡
2q
k/2
j0
k/2
j1
k
2j
2k−2j1
l0
2k − 2j 1 E2k1−2j−l,q El,q
l
3.5
2k−2j1
2k − 2j 1
k
E2k1−2j−l,q El,q
2j − 1
l
l0
⎤
2k − 2j 2k−2j
⎢
q−1
E2k−2j−l,q El,q ⎥
k/2
⎥
k ⎢
l
⎥
⎢
j0
⎥.
⎢
2k−2j1
⎥
⎢
2j
1
q
−
1
2k
−
2j
1
j0
⎣
E2k−2j−l1 El,q ⎦
l
1 q l0
⎡
Equating I3 and I4 , we get the following theorem.
Theorem 3.2. For k ∈ N, one has
k k
l0
l
−1k−l 2q Ekl1,q − qEkl,q
2q
k/2
k
2j
j0
k/2
j1
k
2j − 1
l0
2k − 2j 1 E2k1−2j−l,q El,q
l
2k−2j1
2k − 2j 1
E2k1−2j−l,q El,q
l
l0
⎫
⎧
2k−2j
2k − 2j ⎪
⎪ ⎪
⎪
⎪
E
E
q
−
1
⎪
2k−2j−l,q l,q
⎬
⎨
l
k
j0
.
⎪
2j 1 ⎪
q − 1 2k−2j1
2k − 2j 1 ⎪
⎪
⎪
E2k−2j−l1 El,q ⎪
⎭
⎩
l
1 q l0
k/2
j0
2k−2j1
3.6
Abstract and Applied Analysis
9
Now, we consider 2.8 and 2.1 by means of q-Volkenborn integral. Then, by 2.8,
we see
2q
Zp
xk x − 1m dμq x
m m
m−l
2q
xlk dμq x
−1
l
Z
p
l0
2q
3.7
m m
−1m−l Blk,q .
l
l0
On the other hand,
max{k,m}
j0
km−j
k m − j k
j m
xl dμq x
q
−1
Ekm−j−l,q
j
j
l
Z
p
l0
max{k,m}
j0
km−j
k m − j k
m
q
−1j
Ekm−j−l,q Bl,q .
j
j
l
l0
3.8
Therefore, we get the following theorem.
Theorem 3.3. For m, k ∈ N, one has
2q
m m
l
l0
−1m−l Blk,q
max{k,m}
j0
km−j
k m − j k
m
q
−1j
Ekm−j−l,q Bl,q .
j
j
l
l0
3.9
By using fermionic p-adic q-integral on Zp in left hand side of 2.21, we get
I5 2q
2q
Zp
k k
l0
2q
xk x − 1k 2x − q dμq x
l
k k
l0
l
−1k−l
Zp
xkl1 dμq x − q
−1k−l Bkl1,q − q
k k
l0
k k
l0
l
l
−1k−l
−1k−l Bkl,q .
Zp
xkl dμq x
3.10
10
Abstract and Applied Analysis
Also, we consider right hand side of 2.21 as follows:
I6 2q
k/2
j0
k
2j
2k−2j1
l0
2k − 2j 1 xl dμq x
E2k1−2j−l,q
l
Zp
2k−2j1
2k − 2j 1
k
xl dμq x
E2k1−2j−l,q
2j − 1
l
Zp
l0
k/2
j1
⎤
2k − 2j 2k−2j
⎢
xl dμq x ⎥
q−1
E2k−2j−l,q
k/2
⎥
k ⎢
l
Zp
⎥
⎢
j0
⎥
⎢
2k−2j1
⎥
⎢
2j
1
2k − 2j 1 j0
⎣ q − 1
l
x dμq x⎦
E2k−2j−l1
l
1 q l0
Zp
⎡
2q
k/2
j0
k
2j
2k−2j1
l0
2k − 2j 1 E2k1−2j−l,q Bl,q
l
3.11
k/2
2k−2j1
2k − 2j 1
k
E2k1−2j−l,q Bl,q
2j − 1
l
l0
j1
⎤
2k − 2j 2k−2j
⎢
E2k−2j−l,q Bl,q ⎥
q−1
k/2
⎥
k ⎢
l
⎥
⎢
j0
⎥.
⎢
2k−2j1
⎥
⎢
2j
1
j0
⎦
⎣ q − 1 2k − 2j 1 E
2k−2j−l1 Bl,q
l
1 q l0
⎡
Equating I5 and I6 , we get the following corollary.
Corollary 3.4. For k ∈ N, one gets
k k
l0
l
−1k−l 2q Bkl1,q − qBkl,q
2q
k/2
j0
k/2
j1
k
2j
2k−2j1
l0
2k − 2j 1 E2k1−2j−l,q Bl,q
l
2k−2j1
2k − 2j 1
k
E2k1−2j−l,q Bl,q
2j − 1
l
l0
⎫
⎧
2k − 2j 2k−2j
⎪
⎪
⎪
⎪
⎪
E2k−2j−l,q Bl,q ⎪
q−1
⎪
k/2
⎬
⎨
k ⎪
l
j0
.
⎪
2j 1 ⎪
2k − 2j 1
⎪
⎪
q − 1 2k−2j1
j0
⎪
⎪
⎪
⎪
E2k−2j−l1 Bl,q ⎭
⎩
l
1 q l0
3.12
Abstract and Applied Analysis
11
Acknowledgment
The authors would like to thank the referee for his/her valuable comments on this work.
References
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