Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 371295, 9 pages
doi:10.1155/2008/371295
Research Article
Note on q-Extensions of Euler Numbers and
Polynomials of Higher Order
Taekyun Kim,1 Lee-Chae Jang,2 and Cheon-Seoung Ryoo3
1
The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University,
Taegu 702-701, South Korea
2
Department of Mathematics and Computer Science, KonKuk University,
Chungju 143-701, South Korea
3
Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Correspondence should be addressed to Cheon-Seoung Ryoo, ryoocs@hnu.kr
Received 1 November 2007; Accepted 22 December 2007
Recommended by Paolo Emilio Ricci
In 2007, Ozden et al. constructed generating functions of higher-order twisted h, q-extension of
Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral on Zp . By applying their generating functions, they derived the complete sums of products of the twisted h, qextension of Euler polynomials and numbers. In this paper, we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions
which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore, we
study Barnes-type q-Euler zeta functions. Finally, we will derive the new formula for “sums of products of q-Euler numbers and polynomials” by using fermionic p-adic, q-integral on Zp .
Copyright q 2008 Taekyun Kim 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 and notations
Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational
numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let ν p be the normalized exponential valuation of Cp with |p|p p−νp p p−1 . When
one talks of q-extension, q is considered in many ways such as an indeterminate, a complex
number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C, one normally assumes that |q| < 1. If q ∈ Cp ,
we normally assume that |q − 1|p < p−1/p−1 so that qx exp x log q for |x|p ≤ 1. In this paper,
we use the following notation:
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Journal of Inequalities and Applications
1 − qx
1−q
xq x : q 1.1
cf. 1–5, 22.
Hence, lim q→1 x x for any x with |x|p ≤ 1 in the present p-adic case. Let d be a fixed
integer and let p be a fixed prime number. For any positive integer N, we set
Z
,
dpN Z
N
X∗ a dpZp ,
X lim
←
1.2
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z lies in 0 ≤ a < dpN . For any positive integer N ,
qa
μq a dpN Zp N dp q
1.3
is known to be a distribution on X cf. 1–20. From this distribution, we derive the p-adic,
q-integral on Zp as follows:
1
fxdμq x lim N N→∞ p
Zp
q
N
p
−1
qx fx,
f ∈ UD Zp ,
1.4
x0
see 1–23.
Higher-order twisted Bernoulli and Euler numbers and polynomials are studied by
many authors see for detail 1–21. In 14 Ozden et al. constructed generating functions
of higher-order twisted h, q-extension of Euler polynomials and numbers, by using p-adic,
q-deformed fermionic integral on Zp . By applying their generating functions, they derived the
complete sums of products of the twisted h, q-extension of Euler polynomials and numbers,
see 14, 15. In this paper, we consider the new q-extension of Euler numbers and polynomials
to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials,
we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore, we study
Barnes-type q-Euler zeta functions. Finally, we will derive the new formula for “sums of products of q-Euler numbers and polynomials” by using fermionic p-adic, q-integral on Zp .
2. q-extension of Euler numbers
In this section we assume that q ∈ C with |q| < 1. Now we consider the q-extension of Euler
polynomials as follows:
Fq x, t 2q
qet
1
ext ∞ E x
n,q
n0
n!
tn ,
|t log q| < π.
2.1
Taekyun Kim et al.
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Note that
lim Fq x, t Fx, t et
q→1
∞
En x n
2
ext t .
n!
1
n0
2.2
In the special case x 0, the q-Euler polynomial En,q 0 En,q will be called q-Euler numbers.
It is easy to see that Fq x, t is analytic function in C. Hence we have
∞ E x
n,q
n!
n0
tn 2q
qet 1
ext 2q
∞
−1n qn enxt .
2.3
n0
If we take the kth derivative at t 0 on both sides in 2.3, then we have
Ek,q x 2q
∞
−1n qn n xk .
2.4
n0
From 2.4 we can define q-zeta function which interpolating q-Euler numbers at negative integer as follows.
For s ∈ C, we define
ζq s, x 2q
∞
−1n qn
s,
n0 n x
s ∈ C.
2.5
Note that ζq s, x is analytic in complex s-plane. If we take s −k k ∈ Z , then we have
ζq −k, x Ek,q x.
By 2.4 and 2.5, we obtain the following.
Theorem 2.1. For k ∈ Z ,
Ek,q x 2q
∞
−1n qn n xk .
2.6
n0
Let Fq 0, t Fq t. Then
2q
n−1
−1k qk ekt k0
2q
1 qet
− 2q
−1n qn ent
1 qet
2.7
n n
Fq t − −1 q Fq n, t.
From 2.7, derive
∞
k0
2q
n−1
l0
l l k
−1 q l
∞
tk
tk Ek,q − −1n qn Ek,q n
.
k! k0
k!
By comparing the coefficients on both sides in 2.8, we obtain the following.
2.8
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Journal of Inequalities and Applications
Theorem 2.2. Let n ∈ N, k ∈ Z . If n ≡ 0 mod 2, then
Ek,q − qn Ek,q n 2q
n−1
−1l ql lk .
2.9
−1l ql lk .
2.10
l0
If n ≡ 1 mod 2, then
Ek,q qn Ek,q n 2q
n−1
l0
For w1 , w2 , . . . , wr ∈ C, consider the multiple q-Euler polynomials of Barnes-type as follows:
2rq ext
qew1 t 1 qew2 t 1 · · · qewr t 1
∞
tn
En,q w1 , . . . , wr | x
, where max |wi t log q| < π.
1≤i≤r
n!
n0
Fqr w1 , w2 , . . . , wr | x, t 2.11
For x 0, En,q w1 , . . . , wr | 0 En,q w1 , . . . , wr will be called the multiple q-Euler
numbers of Barnes type. It is easy to see that Fqr w1 , w2 , . . . , wr | x, t is analytic function in the
given region. From 2.11, we derive
2rq
∞
−q
r
i1 ni
e
r
i1 ni wi xt
∞
tn
.
En,q w1 , . . . , wr | x
n!
n0
n1 ,...,nr 0
2.12
By the kth differentiation on both sides in 2.12, we see that
2rq
∞
−q
r
i1 ni
n1 ,...,nr 0
r
k
ni wi x
Ek,q w1 , . . . , wr | x .
2.13
i1
From 2.12, we can derive the following Barnes-type multiple q-Euler zeta function as follows.
For s ∈ C, define
∞
r
ζr,q w1 , w2 , . . . , wr | s, x 2q
n1 ,...,nr 0
−1n1 ···nr qn1 ···nr
n1 w1 n2 w2 · · · nr wr x
s .
2.14
By 2.13 and 2.14, we obtain the following.
Theorem 2.3. For k ∈ Z , w1 , w2 , . . . , wr ∈ C,
ζr,q w1 , w2 , . . . , wr | −k, x Ek,q w1 , w2 , . . . , wr | x .
2.15
Let χ be the primitive Drichlet character with conductor f odd ∈ N. Then we consider
generalized Euler numbers attached to χ as follows:
Fχ,q t 2q
f−1
a a
at
a0 −1 q χae
qf eft 1
∞
n0
En,χ,q
tn
,
n!
2.16
Taekyun Kim et al.
5
where | log q t| < π/f. The numbers En,χ,q will be called the generalized q-Euler numbers
attached to χ. From 2.16, note that
Fχ,q t 2q
2q
f−1
a a
at
a0 −1 q χae
qf eft 1
f−1
∞
−1a qa χa qnf −1n eanft
a0
2q
2q
2.17
n0
f−1
∞ −1anf qanf χa nfeanft
n0 a0
∞
∞
n0
n0
−1n qn χnent En,χ,q
tn
.
n!
Thus,
Ek,χ,q ∞
dk
F
t
2
−1n qn χnnk ,
χ,q
q
dtk
t0
n1
k ∈ N.
2.18
Therefore, we can define the Dirichlet-type l-function which interpolates at negative integer as
follows.
For s ∈ C, we define lq s, χ as
lq s, χ 2q
∞
−1n qn χn
ns
n1
.
2.19
By 2.18 and 2.19, we obtain the following.
Theorem 2.4. For k ∈ Z ,
lq −k, χ Ek,χ,q .
2.20
From 2.1 and the definition of q-Euler numbers, derive
∞
tn xl l
t
n! l0 l!
qet 1
n0
∞
m
tm
m
m−n
En,q
x
.
n
m!
m0 n0
Fq t, x 2q
ext ∞
En,q
2.21
By 2.21 it is shown that
n
Em,q
xn−m ,
En,q x m
m0
n
n ∈ Z .
2.22
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Journal of Inequalities and Applications
For f (=odd) ∈ N, note that
∞
En,q x
n0
2q xt
tn
t
e
n! qe 1
2q
f−1
1
qf eft 1 a0
f−1
2q 2qf
−1a qa eax/fft
a a
−1 q
2qf eax/fft
2.23
qf eft 1
f−1
∞
2q a x f n tn
a a
−1 q
En,qf
.
2qf a0
f
n!
n0
a0
Thus, we have the distribution relation for q-Euler polynomials as follows.
Theorem 2.5. For f (=odd) ∈ N,
En,q x f−1
f n 2q 2qf
−1a qa En,qf
a0
ax
.
f
2.24
For k, n ∈ N with n ≡ 0 (mod 2), it is easy to see that
2q
n−1
−1l−1 ql lk qn Ek,q n − Ek,q
l0
k
nk−m Em,q − Ek,q
q
m
m0
k−1
k
n
Em,q nk−m qn − 1 Ek,q .
q
m
m0
n
k
2.25
Therefore, we obtain the following.
Theorem 2.6. For k, n ∈ N with n ≡ 0 (mod 2),
n−1
k−1
k
l−1 l k
n
2q −1 q l q
Em,q nk−m qn − 1 Ek,q .
m
m0
l0
2.26
3. Witt-type formulae on Zp in p-adic number field
In this section, we assume that q ∈ Cp with |1 − q|p < 1. g is a uniformly differentiable function
at a point a ∈ Zp , and write g ∈ UDZp if the difference quotient
Fg x, y gx − gy
x−y
3.1
has a limit f a as x, y→a, a. For g ∈ UDZp , an invariant p-adic, q-integral is defined as
Iq g 1
Zp
gxdμq x lim N N→∞ p
q
N
−1
p
gxqx .
x0
3.2
Taekyun Kim et al.
7
The fermionic p-adic, q-integral is also defined as
I−q g gxdμ−q x lim
2q p−1
gx−1x qx
N
N→∞ 1 qp
x0
Zp
N
3.3
see 4.
From 3.3, we have the integral equation as follows:
qI−q g1 I−q g 2q g0,
If we take gx e , then we have
g1 x gx 1.
3.4
tx
Iq etx From 3.5, we note that
∞ n0
Zp
Zp
ext dμ−q x xn dμ−q x
2q
qet 1
.
∞
2q
tn
tn
En,q .
t
n! qe 1 n0
n!
By comparing the coefficient on both sides, we see that
xn dμ−q x En,q , n ∈ Z .
Zp
By the same method, we see that
∞
2q xt tn
e exyt dμ−q y t
En,q x .
n!
qe 1
Zp
n0
Hence, we have the formula of Witt’s type for q-Euler polynomial as follows:
x yn dμ−q y En,q x, n ∈ Z .
Zp
3.5
3.6
3.7
3.8
3.9
For n ∈ Z , let gn x gx n. Then we have
n−1
qn I−q gn −1n−1 I−q g 2q −1n−1−l ql gl.
3.10
l0
If n is odd positive integer, then we have
n−1
qn I−q gn I−q g 2q −1l ql gl.
3.11
l0
Let χ be the primitive Drichlet character with conduct f odd ∈ N and let gx χxext . From 3.5 we derive
xt
I−q χxe χxetx dμ−q x
X
2q
∞
n0
f−1
a a
at
a0 −1 q χae
qf eft 1
En,χ,q
3.12
tn
.
n!
Thus, we have the Witt formula for generalized q-Euler numbers attached to χ as follows:
χxxn dμ−q x En,χ,q , n ≥ 0.
3.13
X
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Journal of Inequalities and Applications
4. Higher-order q-Euler numbers and polynomials
In this section we also assume that q ∈ Cp with |1 − q|p < 1. Now we study on higher-order
q-Euler numbers and polynomials and sums of products of q-Euler numbers. First, we try to
consider the multivariate fermionic p-adic, q-integral on Zp as follows:
···
ea1 x1 a2 x2 ···ar xr t ext dμ−q x1 · · · dμ−q xr
Zp
Zp
r times
4.1
2rq
ext ,
qea1 t 1 qea2 t 1 · · · qear t 1
where a1 , a2 , . . . , ar ∈ Zp .
From 4.1 we consider the multiple q-Euler polynomials as follows:
2rq
ext qea1 t 1 qea2 t 1 · · · qear t 1
∞
tn
En,q a1 , a2 , . . . , ar | x
.
n!
n0
In the special case a1 , a2 , . . . , ar 1, 1, . . . , 1, we write
r
En,q a1 , . . . , ar | x En,q x.
4.2
4.3
r times
For x 0, the multiple q-Euler polynomials will be called as q-Euler numbers of order r .
From 4.2 we note that
r
n En,q a1 , a2 , . . . , ar | x ···
dμ−q xj .
a1 x1 · · · ar xr x
4.4
Zp
Zp
j1
r times
It is easy to check that
En,q a1 , a2 , . . . , ar | x n
l0
n
xn−l El,q a1 , a2 , . . . , ar ,
l
4.5
where En,q a1 , a2 , . . . , ar En,q a1 , a2 , . . . , ar | 0. Multinomial theorem is well known as follows:
n
r
r
n
xj
xala ,
4.6
l
,
.
.
.
,
l
1
r
j1
a1
l ,...,l ≥0
1
r
l1 ···lr n
where
n
l1 , . . . , l r
By 4.2 and 4.6 we easily see that
r
En,q x
n
m0 l1 ,...,lr ≥0
l1 ···lr m
n!
.
l1 !l2 ! · · · lr !
r
n
m
Elj ,q .
xn−m
m
l1 , . . . , lr
j1
4.7
4.8
Taekyun Kim et al.
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