fference EquationsVolume 2008, Article ID 738603, 11 pages doi:10.1155/2008/738603 Research Article Lee-Chae Jang Department of Mathematics and Computer Science, Konkuk University, Chungj
Trang 1fference Equations
Volume 2008, Article ID 738603, 11 pages
doi:10.1155/2008/738603
Research Article
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 onZp , 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, Z p,Qp, andCpare, respectively, the
ring of p-adic rational integers, the field of p-adic rational numbers, and the p-adic completion
of the algebraic closure ofQp The p-adic absolute value in C pis 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 ∈ C p If q ∈ C, one normally assumes that |q| < 1 If q ∈ C p, one normally assumes that|1 − q| p < p −1/p−1 so that q x expx log q for each x ∈ Z p We use the notations
x q 1− q x
1− q , x −q
1− −q x
cf 1 14, for all x ∈ Z p For a fixed odd positive integer d with p, d 1, set
X X d lim←
n Z/dp n Z, X1 Zp ,
Trang 2X∗
0<a<dp
a,p1
a dpZ p ,
a dp nZp x ∈ X | x ≡ a
mod dp n
,
1.2
where a ∈ Z lies in 0 ≤ a < dp n For any n ∈ N,
μ q
a dp nZp
dp q a n
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 ∈ Z p and denote this
property by f ∈ UDZ p if the difference quotients
F f x, y fx − fy
have a limit l fa as x, y → a, a cf 25
The p-adic q-integral of a function f ∈ UDZ p was defined as
I q f
Zp
n→∞
1
p n
q
p n−1
x0
I −q f
Zp
n→∞
1
p n
q
p n−1
x0
cf 4,24,25,28, from 1.6, we derive
qI −q
f1
where f1x fx 1 If we take fx e tx , then we have f1x e tx1 e tx e t From1.7,
we obtain that
I −q
e tx
2q
In Section 2, we define the multiple twisted q-Euler numbers and polynomials on Z p and find Witt’s type formula for multiple twisted q-Euler numbers We also have sums of consecutive multiple twisted q-Euler numbers InSection 3, we consider multiple twisted q-Euler Zeta functions which interpolate new multiple twisted q-q-Euler polynomials at negative
integers and investigate some characterizations of them InSection 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
Trang 32 Multiple twistedq-Euler numbers and polynomials
In this section, we assume that q ∈ C p with|1 − q| p < 1 For n ∈ N, by the definition of p-adic
q n I −q
f n
−1n−1 I −q f 2 q n−1
x0
−1n−1−x q x fx, 2.1
where f n x fx n If n is odd positive integer, we have
q n I −q
f n
I −q f 2 q n−1
x0
−1n−1−x q x fx. 2.2
Let T p ∪n≥1 C p n limn→∞ C p n C p∞ be the locally constant space, where C p n {w |
w p n 1} is the cyclic group of order p n For w ∈ T p, we denote the locally constant function by
cf 5,7 14,16,18 If we take fx φ w xe tx, then we have
Zp
e tx φ w xdμ −q x 2q
Now we define the twisted q-Euler numbers E q n,w as follows:
F w t 2q
∞
no
E q n,w t n
We note that by substituting w 1, lim q→1 E q n,1 E nare 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
Zp
x n w x dμ −q x E q
for each w ∈ T p and n ∈ N.
Twisted q-Euler polynomials E q n,w x are defined by means of the generating function
F w q t, x 2q
qwe t 1e xt
∞
n0
E q n,w x t n
where E q n,w 0 E q
n,w By using the hth iterative fermionic p-adic q-integral on Z p, we define
multiple twisted q-Euler number as follows:
Zp
· · ·
Zp
h-times
w x1···x h e x1x2···x h t dμ −q
x1
· · · dμ −q
x h
2q
h
∞
n0
E h,q n,w t n n! . 2.8
Thus we give Witt’s type formula for multiple twisted q-Euler numbers as follows.
Trang 4Theorem 2.1 For each w ∈ T p and h, n ∈ N,
Zp
· · ·
Zp
h-times
w x1···x h
x1 · · · x h
n
dμ −q
x1
· · · dμ −q
x h
E n,w h,q , 2.9
where
x1 · · · x hn
l1···l h n
l1, ,l h≥0
n!
l1!· · · l h!x l1
1 · · · x l h
From2.8 and 2.9, we obtain the following theorem
Theorem 2.2 For w ∈ T p and h, k ∈ N,
E h,q k,w
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!E q l1,w · · · E q
From these formulas, we consider multivariate fermionic p-adic q-integral on Z p as follows:
Zp
· · ·
Zp
h-times
w x1···x h e x1···x h xt dμ −q
x1
· · · dμ −q
x h
2q
· · ·
2q
e xt
2q
h
e xt
2.12
Then we can define the multiple twisted q-Euler polynomials E n,w h,q x as follows:
F w h,q t, x
2q
h
n0
E h,q n,w x t n
From2.12 and 2.13, we note that
∞
n0
Zp
· · ·
Zp
h-times
w x1···x h
x1 · · · x h xn
dμ −q
x1
· · · dμ −q
x h t n
n0
E h,q n,w x t n
n! . 2.14
Then by the kth differentiation on both sides of 2.14, we obtain the following
Theorem 2.3 For each w ∈ T p and k, h ∈ N,
Zp
· · ·
Zp
h-times
w x1···x h
x1 · · · x h xk
dμ −q
x1
· · · dμ −q
x h
E h,q k,w x. 2.15
Trang 5Note that
x1 · · · x h x n
l1···l h n
l1, ,l h≥0
n!
l1!· · · l h!x l1
1 · x l2
2 · · · x h x l h 2.16
Then we see that
Zp
· · ·
Zp
h-times
w x1···x h
x1 · · · x h xk
dμ −q
x1
· · · dμ −q
x h
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!
Zp
w x1x l1
1dμ −q
x1
· · ·
Zp
w x h−1 x l h−1
h−1 dμ −q
x h−1
Zp
x x h
l h
dμ −q
x h
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!E l q1,w · · · E q
l h−1 ,w E q l h ,w x.
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 ∈ T p and k, h ∈ N,
E h,q k,w x
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!E q l1,w · · · E q
l h−1 ,w · E q
3 Multiple twistedq-Euler Zeta functions
For q ∈ C with |q| < 1 and w ∈ T p , the multiple twisted q-Euler numbers can be considered as
follows:
F w h t
q
h
∞
n0
E h,q n,w t n
From3.1, we notethat
∞
n0
E h,q n,w t n n! F h
w t
2q
h
2h q
2q
· · ·
2q
2h q
∞
n1 0
−1n1
q n1w n1e n1t· · · ∞
n h0
−1n h q n h w n h e n h t
2h q
n , ,n0
−1n1···n h q n1···n h w n1···n h e n1···n h t
3.2
Trang 6By the kth differentiation on both sides of 3.2 at t 0, we obtain that
E h,q k,w 2h
q
n1···n h / 0
n1, ,n h≥0
−1n1···n h q n1···n h w n1···n h
n1 · · · n hk
From3.3, we derive multiple twisted q-Euler Zeta function as follows:
ζ h,q w s 2 h
q
n1···n h / 0
n1, ,n h≥0
−1n1···n h q n1···n h w n1···n h
n1 · · · n h
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 ∈ T p and k, h ∈ N,
ζ w h,q −k E h,q k,w 3.5
4 Multiple twisted Barnes’ typeq-Euler polynomials
In this section, we consider the generating function of multiple twisted q-Euler polynomials:
F h
w t, x 2q
h
n0
E n,w h,q x t n
n! ,
We note that
∞
n0
E h,q n,w x t n
n! F h
w t, x 2 h
q
n1, ,n h0
−1n1···n h q n1···n h w n1···n h e n1···n h xt 4.2
By the kth differentiation on both sides of 4.2 at t 0, we obtain that
E h,q k,w x 2 h
q
n1, ,n h0
−1n1···n h q n1···n h w n1···n h n1 · · · n h x k 4.3
Thus we can consider multiple twisted Hurwitz’s type q-Euler Zeta function as follows:
ζ w h,q s, x 2 h
q
n1···n h / 0
n1, ,n h≥0
−1n1···n h q n1···n h w n1···n h
for all s ∈ C and Rex > 0 We note that ζ h,q w s, x is analytic function in the whole complex
s-plane and ζ h,q w s, 0 ζ w h,q s We also remark that if w 1 and h 1, then ζ 1,q1 s, x
ζ q 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
Trang 7Theorem 4.1 For w ∈ T p , k, h ∈ N, s ∈ C, and Rex > 0,
ζ h,q w −k, x E h,q k,w x. 4.5 Let us consider
F w h
a1, , a h | t, x
2q
qwe a1t 1
· · ·
2q
qwe a h t 1
e xt
2h q
∞
n1, ,n h0
−1n1···n h q n1···n h w n1···n h e a1n1···a h n h xt
n0
E n,w h,q
a1, , a h | x t n
n! ,
4.6
where a1, , a h ∈ C and max1≤i≤k{| logq a i t|} < π Then E h,q n,w a1, , a h | x will be called multiple twisted Barnes’ type q-Euler polynomials We note that
E h,q n,w 1, 1, , 1 | x E h,q n,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 ∈ T p , a1, , a h ∈ C, k, h ∈ N, and Rex > 0,
E h,q k,w
a1, , a h | x 2h
q
n1···n h / 0
n1, ,n h≥0
−1n1···n h q n1···n h w n1···n h
a1n1 · · · a h n h xk
,
4.8
where
a1n1 · · · a h n h xk
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!a l1
1 · · · a l h−1
h−1 n l1
1· · · n l h−1
h−1
a h n h xl h
From4.8, we consider multiple twisted Barnes’ type q-Euler Zeta function defined as follows: for each w ∈ T p , a1, , a h ∈ C, k, h ∈ N, and Rex > 0,
ζ h,q k,w
a1, , a h | s, x 2h
q
n1···n h / 0
n1, ,n h≥0
−1n1···n h q n1···n h w n1···n h
a1n1 · · · a h n h xs 4.10
We note that ζ h,q k,w a1, , a h | 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 ∈ T p , a1, , a h ∈ C, k, h ∈ N, and Re x > 0,
ζ h,q k,w
a1, , a h | −k, x E h,q k,w a1, , a h | x. 4.11
Trang 85 Multiple twisted Dirichlet’s typeq-Euler numbers and polynomials
Let χ be a Dirichlet’s character with conductor d odd ∈ N and w ∈ T p If we take fx
X
χxw x e tx dμ −q x 2q
d−1
i0 −1d−1−i q i χiw i e ti
In view of5.1, we can define twisted Dirichlet’s type q-Euler numbers as follows:
F w,χ q t 2q
d−1
i0 −1d−1−i q i χiw i e ti
∞
n0
E q n,χ,w t n n! ,t logqw< π
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 ∈ T p ,
X χxw x e tx dμ −q x E q
n,χ,w 5.3
We note that if w 1, then E q n,χ,1 E q n,χ is the generalized q-Euler numbers attached to χ
see 18,26 From 5.2, we also see that
F w,χ q t 2 q d−1
i0
−1d−1−i q i χiw i e ti
∞
l0
q ld w ld e ldt−1l
2q ∞
n0
−1n q n w n χne nt
5.4
By5.2 and 5.4, we obtain that
E k,χ,w q d k
dt k F w,χ q t | t0 2q ∞
n0
From5.5, we can define the l q
w,χ-function as follows:
l q χ,w s 2 q ∞
n0
−1n q n w n χn
for all s ∈ C We note that l χ,w q s is analytic function in the whole complex s-plane From 5.5 and5.6, we can derive the following result
Theorem 5.2 Let χ be a Dirichlet’s character with conductor d odd ∈ N For each w ∈ T p ,
l w,χ q −n E q
Trang 9Now, in view of5.1, we can define multiple twisted Dirichlet’s type q-Euler numbers
by means of the generating function as follows:
F w,χ h,q t
2qd−1 i0 −1d−1−i q i χiw i e ti
q d w d e td 1
h
X
χxw x e tx dμ −q x
h
∞
n0
E h,q n,χ,w t n n! ,
5.8 where|t logqw| < π/d We note that if w 1, then E q
n,χ,1 is a multiple generalized q-Euler
numbersee 22
By using the same method used in2.8 and 2.9,
∞
n0
X
· · ·
X
h-times
χ
x1 · · · x hw x1···x h
x1 · · · x hn
dμ −q
x1
· · · dμ −q
x h t n
n0
E n,w h,q t n n! .
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 ∈ T p , h ∈ N,
X
· · ·
X
h-times
χ
x1 · · · x hw x1···x h
x1 · · · x hn
dμ −q
x1
· · · dμ −qx h
E h,q n,χ,w , 5.10
where χx1 · · · x h χx1 · · · χx h and
x1 · · · x hn
l1···l h n
l1, ,l h≥0
n!
l1!· · · l h!x l1
1 · · · x l h
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 ∈ T p , h ∈ N,
E k,χ,w h,q
l1···l h k
l1, ,lh≥0
k!
l1!· · · l h!E q l1,χ,w · · · E q
Finally, we consider multiple twisted Dirichlet’s type q-Euler polynomials defined by
means of the generating functions as follows:
F w,χ q t, x
2qd−1 i0 −1d−1−i q i χiw i e ti
q d w d e td 1
h
n0
E h,q n,χ,w x t n
Trang 10where|t logqw| < π/d and Rex > 0 From 5.13, we note that
∞
n0
X
· · ·
X
h-times
χ
x1· · ·x hw x1···x h
x1· · · x h xn
dμ −q
x1
· · · dμ −q
x h t n
n0
E h,q n,χ,w x t n
n! .
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 ∈ T p , h ∈ N,
X
· · ·
X
h-times
χ
x1 · · · x hw x1···x h x1 · · · x h x n dμ −q
x1
· · · dμ −q
x h
E h,q n,χ,w x, 5.15
where
x1 · · · x h xn
l1···l h n
l1, ,l h≥0
n!
l1!· · · l h!x l1
1 · · ·x h xl h
Theorem 5.6 Let χ be a Dirichlet’s character with conductor d odd ∈ N For each w ∈ T p , h ∈ N,
E h,q k,χ,w x
l1···l h k
l1, ,l h≥0
k!
l1!· · · l h!E q l1,χ,w · · · E q
l h−1 ,χ,w · E q
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