Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 738603, 11 pages
doi:10.1155/2008/738603
Research Article
Multiple Twisted q-Euler Numbers and Polynomials
Associated with p-Adic q-Integrals
Lee-Chae Jang
Department of Mathematics and Computer Science, Konkuk University,
Chungju 380701, South Korea
Correspondence should be addressed to Lee-Chae Jang, leechae.jang@kku.ac.kr
Received 14 January 2008; Revised 25 February 2008; Accepted 26 February 2008
Recommended by Martin Bohner
By using p-adic q-integrals on Zp , we define multiple twisted q-Euler numbers and polynomials.
We also find Witt’s type formula for multiple twisted q-Euler numbers and discuss some
characterizations of multiple twisted q-Euler Zeta functions. In particular, we construct multiple
twisted Barnes’ type q-Euler polynomials and multiple twisted Barnes’ type q-Euler Zeta functions.
Finally, we define multiple twisted Dirichlet’s type q-Euler numbers and polynomials, and give
Witt’s type formula for them.
Copyright q 2008 Lee-Chae 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.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp are, 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 about 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 that |q| < 1. If q ∈ Cp ,
one normally assumes that |1 − q|p < p−1/p−1 so that qx expx log q for each x ∈ Zp . We use
the notations
xq 1 − qx
,
1−q
x−q 1 − −qx
1q
cf. 1–14, for all x ∈ Zp . For a fixed odd positive integer d with p, d 1, set
X Xd lim
Z/dpn Z,
←
n
X1 Zp ,
1.1
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Advances in Difference Equations
X∗ a dpZp ,
0<a<dp
a,p1
a dpn Zp x ∈ X | x ≡ a mod dpn ,
1.2
where a ∈ Z lies in 0 ≤ a < dpn . For any n ∈ N,
μq a dpn Zp qa
dpn q
1.3
is known to be a distribution on X cf. 1–28.
We say that f is uniformly differentiable function at a point a ∈ Zp and denote this
property by f ∈ UDZp if the difference quotients
Ff x, y fx − fy
x−y
1.4
have a limit l f a as x, y → a, a cf. 25.
The p-adic q-integral of a function f ∈ UDZp was defined as
p −1
1 Iq f fxdμq x lim n fxqx ,
n→∞ p
Zp
q x0
n
p −1
1 fxdμ−q x lim n fx−qx ,
I−q f n→∞ p
Zp
q x0
1.5
n
1.6
cf. 4, 24, 25, 28, from 1.6, we derive
qI−q f1 I−q f 2q f0,
1.7
where f1 x fx 1. If we take fx etx , then we have f1 x etx1 etx et . From 1.7,
we obtain that
2q
I−q etx t
.
qe 1
1.8
In Section 2, we define the multiple twisted q-Euler numbers and polynomials on Zp
and find Witt’s type formula for multiple twisted q-Euler numbers. We also have sums of
consecutive multiple twisted q-Euler numbers. In Section 3, we consider multiple twisted qEuler Zeta functions which interpolate new multiple twisted q-Euler polynomials at negative
integers and investigate some characterizations of them. In Section 4, we construct the multiple
twisted Barnes’ type q-Euler polynomials and multiple twisted Barnes’ type q-Euler Zeta
functions which interpolate new multiple twisted Barnes’ type q-Euler polynomials at negative
integers. In Section 5, we define multiple twisted Dirichlet’s type q-Euler numbers and
polynomials and give Witt’s type formula for them.
Lee-Chae Jang
3
2. Multiple twisted q-Euler numbers and polynomials
In this section, we assume that q ∈ Cp with |1 − q|p < 1. For n ∈ N, by the definition of p-adic
q-integral on Zp , we have
n−1
qn I−q fn −1n−1 I−q f 2q
−1n−1−x qx fx,
2.1
x0
where fn x fx n. If n is odd positive integer, we have
n−1
qn I−q fn I−q f 2q
−1n−1−x qx fx.
2.2
x0
Let Tp ∪n≥1 Cpn limn→∞ Cpn Cp∞ be the locally constant space, where Cpn {w |
n
wp 1} is the cyclic group of order pn . For w ∈ Tp , we denote the locally constant function by
φw : Zp −→ Cp , x −→ wx ,
2.3
cf. 5, 7–14, 16, 18. If we take fx φw xetx , then we have
Zp
2q
etx φw xdμ−q x qwet 1
2.4
.
q
Now we define the twisted q-Euler numbers En,w as follows:
Fw t 2q
qwet 1
∞
q
En,w
no
tn
.
n!
2.5
q
We note that by substituting w 1, limq→1 En,1 En are the familiar Euler numbers. Over five
decades ago, Carlitz defined q-extension of Euler numbers cf. 15. From 2.4 and 2.5, we
note that Witt’s type formula for a twisted q-Euler number is given by
q
xn wx dμ−q x En,w .
2.6
Zp
for each w ∈ Tp and n ∈ N.
q
Twisted q-Euler polynomials En,w x are defined by means of the generating function
q
Fw t, x q
2q
qwet 1
ext ∞
q
En,w x
n0
tn
,
n!
2.7
q
where En,w 0 En,w . By using the hth iterative fermionic p-adic q-integral on Zp , we define
multiple twisted q-Euler number as follows:
Zp
···
Zp
wx1 ···xh ex1 x2 ···xh t dμ−q x1 · · · dμ−q xh 2q
qwet
1
h
∞
n
h,q t
En,w
n0
h-times
Thus we give Witt’s type formula for multiple twisted q-Euler numbers as follows.
n!
.
2.8
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Advances in Difference Equations
Theorem 2.1. For each w ∈ Tp and h, n ∈ N,
Zp
···
Zp
n
h,q
wx1 ···xh x1 · · · xh dμ−q x1 · · · dμ−q xh En,w ,
2.9
h-times
where
x1 · · · xh
n
l1 ···lh n
l1 ,...,lh ≥0
n!
xl1 · · · xhlh .
l1 ! · · · l h ! 1
2.10
From 2.8 and 2.9, we obtain the following theorem.
Theorem 2.2. For w ∈ Tp and h, k ∈ N,
h,q
Ek,w l1 ···lh k
l1 ,...,lh ≥0
k!
q
q
E · · · Elh ,w .
l1 ! · · · lh ! l1 ,w
2.11
From these formulas, we consider multivariate fermionic p-adic q-integral on Zp as
follows:
2q
2q
x1 ···xh x1 ···xh xt
··· w
e
dμ−q x1 · · · dμ−q xh ···
ext
t1
t1
qwe
qwe
Zp
Zp
2.12
h-times
h
2q
ext .
qwet 1
h,q
Then we can define the multiple twisted q-Euler polynomials En,w x as follows:
h,q
Fw t, x
2q
qwet 1
h
ext ∞
n0
h,q
En,w x
tn
.
n!
2.13
From 2.12 and 2.13, we note that
∞ n0
∞
n
tn tn
h,q
wx1 ···xh x1 · · · xh x dμ−q x1 · · · dμ−q xh
En,w x .
n! n0
n!
Zp
Zp
···
2.14
h-times
Then by the kth differentiation on both sides of 2.14, we obtain the following.
Theorem 2.3. For each w ∈ Tp and k, h ∈ N,
Zp
···
Zp
h-times
k
h,q
wx1 ···xh x1 · · · xh x dμ−q x1 · · · dμ−q xh Ek,w x.
2.15
Lee-Chae Jang
5
Note that
x1 · · · xh xn n!
xl1 · xl2 · · · xh xlh .
l1 ! · · · lh ! 1 2
l1 ···lh n
l1 ,...,lh ≥0
2.16
Then we see that
Zp
···
Zp
k
wx1 ···xh x1 · · · xh x dμ−q x1 · · · dμ−q xh
h-times
l1 ···lh k
l1 ,...,lh ≥0
l1 ···lh k
l1 ,...,lh ≥0
k!
l1 ! · · · l h !
Zp
wx1 x1l1 dμ−q x1 · · ·
Zp
lh−1
wxh−1 xh−1
dμ−q xh−1
Zp
x xh
lh
dμ−q xh
k!
q
q
q
E · · · Elh−1 ,w Elh ,w x.
l1 ! · · · lh ! l1 ,w
2.17
From 2.15 and 2.17, we obtain the sums of powers of consecutive q-Euler numbers as
follows.
Theorem 2.4. For each w ∈ Tp and k, h ∈ N,
h,q
Ek,w x l1 ···lh k
l1 ,...,lh ≥0
k!
q
q
q
E · · · Elh−1 ,w · Elh ,w x.
l1 ! · · · lh ! l1 ,w
2.18
3. Multiple twisted q-Euler Zeta functions
For q ∈ C with |q| < 1 and w ∈ Tp , the multiple twisted q-Euler numbers can be considered as
follows:
h
Fw
t
h
2q
qwet
1
∞
n
h,q t
En,w
n0
n!
t logqw < π.
,
3.1
From 3.1, we notethat
∞
n0
n
h,q t
En,w
n!
h
t Fw
2hq
2hq
∞
2q
qwet 1
h
2hq
−1n1 qn1 wn1 en1 t · · ·
n1 0
n1 ,...,nh 0
2q
qwet 1
∞
···
2q
qwet 1
−1nh qnh wnh enh t
nh 0
−1n1 ···nh qn1 ···nh wn1 ···nh en1 ···nh t .
3.2
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Advances in Difference Equations
By the kth differentiation on both sides of 3.2 at t 0, we obtain that
h,q
Ek,w 2hq
k
−1n1 ···nh qn1 ···nh wn1 ···nh n1 · · · nh .
3.3
0
n1 ···nh /
n1 ,...,nh ≥0
From 3.3, we derive multiple twisted q-Euler Zeta function as follows:
h,q
ζw s 2hq
−1n1 ···nh qn1 ···nh wn1 ···nh
s
n1 · · · nh
0
n1 ···nh /
n1 ,...,nh ≥0
3.4
for all s ∈ C. We also obtain the following theorem in which multiple twisted q-Euler Zeta
functions interpolate multiple twisted q-Euler polynomials.
Theorem 3.1. For w ∈ Tp and k, h ∈ N,
h,q
h,q
ζw −k Ek,w .
3.5
4. Multiple twisted Barnes’ type q-Euler polynomials
In this section, we consider the generating function of multiple twisted q-Euler polynomials:
h
Fw
t, x h
2q
qwet
1
t logqw < π,
ext ∞
h,q
En,w x
n0
tn
,
n!
4.1
Rex > 0.
We note that
∞
n0
h,q
En,w x
tn
h
t, x 2hq
Fw
n!
−1n1 ···nh qn1 ···nh wn1 ···nh en1 ···nh xt .
4.2
n1 ,...,nh 0
By the kth differentiation on both sides of 4.2 at t 0, we obtain that
h,q
Ek,w x 2hq
−1n1 ···nh qn1 ···nh wn1 ···nh n1 · · · nh xk .
4.3
n1 ,...,nh 0
Thus we can consider multiple twisted Hurwitz’s type q-Euler Zeta function as follows:
h,q
ζw s, x 2hq
0
n1 ···nh /
n1 ,...,nh ≥0
h,q
−1n1 ···nh qn1 ···nh wn1 ···nh
s
n1 · · · nh x
4.4
for all s ∈ C and Rex > 0. We note that ζw s, x is analytic function in the whole complex
h,q
h,q
1,q
s-plane and ζw s, 0 ζw s. We also remark that if w 1 and h 1, then ζ1 s, x q
ζ s, x is Hurwitz’s type q-Euler Zeta function see 7, 27. The following theorem means
that multiple twisted q-Euler Zeta functions interpolate multiple twisted q-Euler polynomials
at negative integers.
Lee-Chae Jang
7
Theorem 4.1. For w ∈ Tp , k, h ∈ N, s ∈ C, and Rex > 0,
h,q
h,q
ζw −k, x Ek,w x.
4.5
Let us consider
h
Fw
a1 , . . . , ah | t, x ∞
n0
∞
···
qwea1 t 1
2hq
2q
2q
qweah t 1
ext
−1n1 ···nh qn1 ···nh wn1 ···nh ea1 n1 ···ah nh xt
4.6
n1 ,...,nh 0
tn
h,q En,w a1 , . . . , ah | x
,
n!
h,q
where a1 , . . . , ah ∈ C and max1≤i≤k {| logq ai t|} < π. Then En,w a1 , . . . , ah | x will be called
multiple twisted Barnes’ type q-Euler polynomials. We note that
h,q
h,q
En,w 1, 1, . . . , 1 | x En,w x.
4.7
By the kth differentiation of both sides of 4.6, we obtain the following theorem.
Theorem 4.2. For each w ∈ Tp , a1 , . . . , ah ∈ C, k, h ∈ N, and Rex > 0,
h,q Ek,w a1 , . . . , ah | x 2hq
k
−1n1 ···nh qn1 ···nh wn1 ···nh a1 n1 · · · ah nh x ,
0
n1 ···nh /
n1 ,...,nh ≥0
4.8
where
a1 n1 · · · ah nh x
k
l1 ···lh k
l1 ,...,lh ≥0
l
k!
h−1
h−1
nl11 · · · nlh−1
ah nh x h .
al11 · · · alh−1
l1 ! · · · lh !
4.9
From 4.8, we consider multiple twisted Barnes’ type q-Euler Zeta function defined as
follows: for each w ∈ Tp , a1 , . . . , ah ∈ C, k, h ∈ N, and Rex > 0,
h,q ζk,w a1 , . . . , ah | s, x 2hq
n1 ···nh /0
n1 ,...,nh ≥0
−1n1 ···nh qn1 ···nh wn1 ···nh
s .
a1 n1 · · · ah nh x
4.10
h,q
We note that ζk,w a1 , . . . , ah | s, x is analytic function in the whole complex s-plane. We
also see that multiple twisted Barnes’ type q-Euler Zeta functions interpolate multiple twisted
Barnes’ type q-Euler polynomials at negative integers as follows.
Theorem 4.3. For each w ∈ Tp , a1 , . . . , ah ∈ C, k, h ∈ N, and Re x > 0,
h,q h,q ζk,w a1 , . . . , ah | −k, x Ek,w a1 , . . . , ah | x .
4.11
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Advances in Difference Equations
5. Multiple twisted Dirichlet’s type q-Euler numbers and polynomials
Let χ be a Dirichlet’s character with conductor d odd ∈ N and w ∈ Tp . If we take fx χxφw xetx , then we have fd x fx d χxwd etd wx etx . From 2.2, we derive
χxw e dμ−q x x tx
2q
d−1
i0
−1d−1−i qi χiwi eti
qd wd etd 1
X
.
5.1
In view of 5.1, we can define twisted Dirichlet’s type q-Euler numbers as follows:
q
Fw,χ t
2q
d−1
i0
−1d−1−i qi χiwi eti
qd wd etd 1
∞
q
En,χ,w
n0
π
tn , t logqw < ,
n!
d
5.2
cf. 17, 19, 21, 22. From 5.1 and 5.2, we can give Witt’s type formula for twisted Dirichlet’s
type q-Euler numbers as follows.
Theorem 5.1. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp ,
n ∈ N ∪ {0}, we have
q
χxwx etx dμ−q x En,χ,w .
5.3
X
q
q
We note that if w 1, then En,χ,1 En,χ is the generalized q-Euler numbers attached to χ
see 18, 26. From 5.2, we also see that
q
Fw,χ t 2q
d−1
−1d−1−i qi χiwi eti
i0
2q
∞
∞
qld wld eldt −1l
l0
n n
5.4
−1 q w χne .
n
nt
n0
By 5.2 and 5.4, we obtain that
q
Ek,χ,w ∞
dk q
F
t
|
2
−1n qn wn χnnk .
w,χ
t0
q
dtk
n0
5.5
q
From 5.5, we can define the lw,χ -function as follows:
q
lχ,w s 2q
∞
−1n qn wn χn
ns
n0
5.6
q
for all s ∈ C. We note that lχ,w s is analytic function in the whole complex s-plane. From 5.5
and 5.6, we can derive the following result.
Theorem 5.2. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp ,
n ∈ N ∪ {0}, we have
q
q
lw,χ −n En,χ,w .
5.7
Lee-Chae Jang
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Now, in view of 5.1, we can define multiple twisted Dirichlet’s type q-Euler numbers
by means of the generating function as follows:
h,q
Fw,χ t
2q
d−1
i0
−1d−1−i qi χiwi eti
qd wd etd
1
h
h
χxwx etx dμ−q x
X
∞
h,q
En,χ,w
n0
tn
,
n!
5.8
q
where |t logqw| < π/d. We note that if w 1, then En,χ,1 is a multiple generalized q-Euler
number see 22.
By using the same method used in 2.8 and 2.9,
∞ n0
∞
n
n
tn h,q t
· · · χ x1 · · · xh wx1 ···xh x1 · · · xh dμ−q x1 · · · dμ−q xh
En,w .
n! n0
n!
X X
h-times
5.9
From 5.9, we can give Witt’s type formula for multiple twisted Dirichlet’s type q-Euler
numbers.
Theorem 5.3. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp , h ∈ N,
and n ∈ N ∪ {0}, we have
n
h,q
· · · χ x1 · · · xh wx1 ···xh x1 · · · xh dμ−q x1 · · · dμ−q xh En,χ,w ,
X X
5.10
h-times
where χx1 · · · xh χx1 · · · χxh and
x1 · · · x h
n
l1 ···lh n
l1 ,...,lh ≥0
n!
xl1 · · · xhlh .
l1 ! · · · l h ! 1
5.11
From 5.10, we also obtain the sums of powers of consecutive multiple twisted
Dirichlet’s type q-Euler numbers as follows.
Theorem 5.4. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp , h ∈ N,
and n ∈ N ∪ {0}, we have
h,q
Ek,χ,w l1 ···lh k
k!
q
q
E
· · · Elh ,χ,w .
l1 ! · · · lh ! l1 ,χ,w
5.12
l1 ,...,lh ≥0
Finally, we consider multiple twisted Dirichlet’s type q-Euler polynomials defined by
means of the generating functions as follows:
q
Fw,χ t, x
2q
d−1
i0
−1d−1−i qi χiwi eti
qd wd etd
1
h
ext ∞
n0
h,q
En,χ,w x
tn
,
n!
5.13
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Advances in Difference Equations
where |t logqw| < π/d and Rex > 0. From 5.13, we note that
∞ n0
∞
n
tn tn
h,q
· · · χ x1 · · ·xh wx1···xh x1 · · · xh x dμ−q x1 · · · dμ−q xh
En,χ,w x .
n! n0
n!
X X
h-times
5.14
Clearly, we obtain the following two theorems.
Theorem 5.5. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp , h ∈ N,
n ∈ N ∪ {0}, and Rex > 0, we have
h,q
· · · χ x1 · · · xh wx1 ···xh x1 · · · xh xn dμ−q x1 · · · dμ−q xh En,χ,w
x, 5.15
X X
h-times
where
x1 · · · x h x
n
l1 ···lh n
l1 ,...,lh ≥0
l
n!
x1l1 · · · xh x h .
l1 ! · · · l h !
5.16
Theorem 5.6. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ Tp , h ∈ N,
n ∈ N ∪ {0}, and Rex > 0, we have
h,q
Ek,χ,w x l1 ···lh k
l1 ,...,lh ≥0
k!
q
q
q
· · · Elh−1 ,χ,w · Elh ,χ,w x.
E
l1 ! · · · lh ! l1 ,χ,w
5.17
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