Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 793297, 14 pages
doi:10.1155/2008/793297
Research Article
On Multiple Twisted p-adic q-Euler
ζ-Functions and l-Functions
Min-Soo Kim,1 Taekyun Kim,2 and Jin-Woo Son3
1
National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu,
Daejeon 305-340, South Korea
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
3
Department of Mathematics, Kyungnam University, Masan 631-701, South Korea
Correspondence should be addressed to Min-Soo Kim, mskim@kyungnam.ac.kr
Received 6 May 2008; Accepted 13 August 2008
Recommended by Agacik Zafer
We give the existence of multiple twisted p-adic q-Euler ζ-functions and l-functions, which are
generalization of the twisted p-adic h, q-zeta functions and twisted p-adic h, q-Euler l-functions
in the work of Ozden and Simsek 2008.
Copyright q 2008 Min-Soo 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, definitions, and notations
The constants En in the Taylor series expansion
et
∞
2
tn
En
1 n0 n!
1.1
are known as the first kind Euler numbers cf. 1. From the generating function of the first
kind Euler numbers, we note that E0 1 and En − nk0 nk Ek for n ∈ N. The first few
are 1, −1/2, 0, 1/4, −1/2, . . . and E2k 0 for k ∈ N. Those numbers play an important role in
number theory. For example, the Euler zeta-function essentially equals an Euler numbers at
nonpositive integer:
ζE −m Em
for m ≥ 0,
1.2
where
ζE s ∞
−1n
n1
see 1–10.
ns
,
s∈C
1.3
2
Abstract and Applied Analysis
Throughout this paper Z, Zp , Qp , and Cp will denote the ring of integers, the ring
of p-adic rational integers, the field of p-adic rational numbers, and the completion of the
algebraic closure of Qp , respectively. Let vp be the normalized exponential valuation of Cp
with |p|p 1/p. When one talks of q-extension, q is variously considered as an indeterminate,
a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ Cp , then we normally assume
|1 − q|p < 1, so that qx expx log q for x ∈ Zp . If q ∈ C, then we assume that |q| < 1. Also we
use the following notations:
xq 1 − qx
,
1−q
x−q 1 − −qx
,
1q
1.4
cf. 2–4. For
f ∈ UD Zp , Cp f | f : Zp → Cp is uniformly differentiable function ,
1.5
the p-adic q-integral on Zp was defined by Kim cf. 2–4 as follows:
dp
−1
1
fadμq a lim
faqa
Iq f N→∞ dpN Zp
q a0
N
for |1 − q|p < 1.
1.6
Furthermore, we can consider
the fermionic integral in contrast to the conventional bosonic
integral. That is, I−1 f Zp fadμ−1 a cf. 5. From this, we derive
I−1 f1 I−1 f 2f0,
1.7
where f1 a fa 1. Substitute fa ξa qαa eat into 1.7. The twisted α, q-extension of
Euler numbers is defined by 8
I−1 ξa qαa eat ∞
2
tn
α
En,ξ q .
α
t
n!
ξq e 1 n0
1.8
For |1 − q|p < 1, we consider fermionic p-adic q-integral on Zp which is the q-extension of
I−1 f as follows:
I−q f dp−1
1
fa−qa
N→∞ dpN −q a0
N
Zp
fadμ−q a lim
1.9
cf. 5. From 1.9, we can derive the following formula 5:
qI−q f1 I−q f 2q f0,
1.10
where f1 a is translation with f1 a fa 1. If we take fa ξa eat , then we have f1 a ξa1 ea1t ξa eat ξet . From 1.10, we derive ξqet 1I−q ξa eat 2q . Hence, we obtain
I−q ξa eat Zp
ξa eat dμ−q a 2q
ξqet 1
.
1.11
Min-Soo Kim et al.
3
By 1.11, we define the twisted q-Euler numbers, En,q,ξ by means of the following generating
function cf. 5:
2q
ξqet 1
∞
tn
En,q,ξ .
n!
n0
1.12
These numbers are interpolated by the twisted Euler q-zeta function which is defined as
follows:
ζq,ξ,E s 2q
∞
−1n ξn qn
n1
ns
,
s ∈ C.
1.13
Note that ζq,ξ,E s is analytic function in the whole complex plane C.
In view of the functional equation for the twisted Euler q-zeta function at nonpositive
integers, we have
ζq,ξ,E −m Em,q,ξ
for m ≥ 0
1.14
cf. 5.
Twisted q-Bernoulli and Euler numbers and polynomials are very important not
only in practically every field of mathematics, in particular in combinatorial theory, finite
difference calculus, numerical analysis, numbers theory, but also probability theory. Recently
the q-extensions of those Euler numbers polynomials and the multiple of q-extensions of
those Euler numbers polynomials have been studied by many authors, cf. 1–15. In 8,
Ozden and Simsek have studied h, q-extensions of twisted Euler numbers and polynomials
by using p-adic q-integral on the ring of p-adic integers Zp . From their h, q-extensions
of twisted Euler numbers and polynomials, they have derived p-adic h, q-extensions of
Euler zeta function and p-adic h, q-extensions of Euler l-functions. They also gave some
interesting relations between their h, q-Euler numbers and h, q-Euler zeta functions,
and found the p-adic twisted interpolation function of the generalized twisted h, q-Euler
numbers. In 11, Jang defined twisted q-Euler numbers and polynomials of higher order, and
studied multiple twisted q-Euler zeta functions. He also derived sums of products of q-Euler
numbers and polynomials by using fermionic p-adic q-integral. In 7, 9, Ozden et al. defined
multivariate Barnes-type Hurwitz q-Euler zeta functions and l-functions. They also gave
relation between multivariate Barnes-type Hurwitz q-Euler zeta functions and multivariate qEuler l-functions. In 16, Kim constructed multiple p-adic L-functions, which interpolate the
Bernoulli numbers of higher order. He also derived that the values of the partial derivative
of this multiple p-adic L-function at s 0 are given.
In this paper, we consider twisted q-Euler numbers and polynomials of higher
order, and study multiple twisted p-adic, q-Euler, ζ-functions, and l-functions, which are
generalization of the twisted p-adic h, q-zeta functions and twisted p-adic h, q-Euler lfunctions in 8.
2. Preliminaries
We assume that q ∈ C with |q| < 1. Let ξ be a primitive r th root of unity.
4
Abstract and Applied Analysis
For an integer h, the twisted q-Euler polynomials of higher order the index h may be
h
negative, En,q,ξ x, are defined by means of the following generating function cf. 11, 14:
h
Fq,ξ t, x 2q
1
ξqet
2q xt
···
e
1 ξqet
h-times
2hq etx
2hq
∞
n0
∞
−ξl1 ql1 el1 t · · ·
l1 0
∞
∞
−ξlh q lh e lh t
2.1
lh 0
−ξql1 ···lh el1 ···lh xt
l1 ,...,lh 0
h
En,q,ξ x
tn
,
n!
where |t logξq| < π. Note that 2q 1 q, so 2q /1 ξqet ≡ 1 q/1 ξq mod t.
Of course the explicit formulas in 2.1 depend on h which is a positive integer. If h 1, q qα in the above, we obtain the generating function of the twisted α, q-extension of Euler
polynomials in 8, cf. Section 1, 1.3. In fact, if h > 0 then −h < 0. Therefore, the generating
−h
function Fq,ξ t, x is the form
−h
Fq,ξ t, x 2 −h
q
1
ξqet
etx 1 ξqet
2q
h
h
etx ∞
n0
−h
En,q,ξ x
tn
.
n!
2.2
h
The twisted q-Euler numbers of higher order are En,q,ξ En,q,ξ 0. Then, it is immediate that
h
En,q,ξ x
n n
k0
k
h
Ek,q,ξ xn−k .
2.3
From now on, we assume h > 0 and in general whenever h is actually an index then
h > 0. Jang 11 defined the two-variable multiple twisted q-Euler zeta functions as follows.
Definition 2.1. For s ∈ C and x ∈ R {x ∈ R | x > 0}, one defines
h
ζq,ξ,E s, x 2hq
∞
−ξl1 ···lh ql1 ···lh
s .
l1 ,...,lh 0 l1 · · · lh x
2.4
h
ζq,ξ,E s, x is an analytical function in the whole complex plane.
h
The value of ζq,ξ,E s, x at nonpositive integers, Z N ∪ {0}, is given explicitly as
follows.
h
h
Theorem 2.2 see 11. Let m ∈ Z . Then, ζq,ξ,E −m, x Em,q,ξ x.
Min-Soo Kim et al.
5
Let χ be a Dirichlet character with odd conductor d. We define a twisted Dirichlet’s
type q-Euler polynomials of higher order by means of the following generating function cf.
11, 14:
h
Fq,ξ,χ t, x
d−1
1
χ a1 · · · ah −ξqa1 ···ah ea1 ···ah t
dh−q a1 ,...,ah 0
1 qd
1 qd
·
·
·
ext
1 ξd qd edt
1 ξd qd edt
h-times
2hq
d−1
∞
∞
d d dt x1
d d dt xh xt
χ a1 · · · ah −ξqa1 ···ah ea1 ···ah t
···
e
−ξ q e
−ξ q e
a1 ,...,ah 0
x1 0
xh 0
h-times
2hq
2hq
∞
n0
∞
d−1
χ a1 dx1 · · · ah dxh −ξqa1 dx1 ···ah dxh ea1 dx1 ···ah dxh t ext
x1 ,...,xh 0 a1 ,...,ah 0
∞
χ l1 · · · lh −ξql1 ···lh exl1 ···lh t
l1 ,...,lh 0
h
En,q,ξ,χ x
tn
.
n!
2.5
We now see that the twisted Dirichlet’s type q-Euler polynomials of higher order are
easily expressed by the twisted q-Euler polynomials of higher order as follows.
Proposition 2.3. Let F be an odd multiple of the conductor d. Then,
F−1
1
h
En,q,ξ,χ x F n
h
χ a1 · · · ah −ξqa1 ···ah En,qF ,ξF
Fh−q a1 ,...,ah 0
a1 · · · ah x
.
F
2.6
Proof. Let d odd ∈ N. By 2.1 and 2.5, we note that
a1 · · · ah x
h
χ a1 · · · ah −ξqa1 ···ah Fqd ,ξd dt,
.
d
a1 ,...,ah 0
2.7
a1 · · · ah x
a1 ···ah h
χ a1 · · · ah −ξq
En,qd ,ξd
.
d
a1 ,...,ah 0
2.8
d−1
1
h
Fq,ξ,χ t, x dh−q
Then, we have
h
En,q,ξ,χ x
d
n
1
dh−q
d−1
6
Abstract and Applied Analysis
On the other hand, if F dp, then we get
a1 · · · ah x
h
χ a1 · · · ah −ξqa1 ···ah FqF ,ξF Ft,
F
a1 ,...,ah 0
F−1
1
Fh−q
F−1
χ a1 · · · ah −ξqa1 ···ah
a1 ,...,ah 0
d−1
p−1
h
2q
ξF qF eFt 1
ea1 ···ah xt
χ b1 c1 d · · · bh ch d −ξqb1 c1 d···bh ch d
2.9
b1 ,...,bh 0 c1 ,...,ch 0
×
h
2q
ξF qF eFt 1
d−1
1
dh−q
b1 ,...,bh
eb1 c1 d···bh ch dxt
b1 · · · bh x
h
.
χ b1 · · · bh −ξqb1 ···bh Fqd ,ξd dt,
d
0
This completes the proof.
The two-variable multiple twisted q-Euler l-functions are defined by the following
definition.
Definition 2.4 see 14. Let χ be a Dirichlet character. For s ∈ C and x ∈ R , one has
h
li li li
∞
χ l1 · · · lh
h
i1 −1 ξ q
h
lq,ξ,E s, x, χ 2q
.
s
l1 · · · lh x
l1 ,...,lh 0
2.10
h
The value of lq,ξ,E s, x, χ at nonpositive integers is given explicitly by the following
theorem.
h
h
Theorem 2.5 see 14. Let m ∈ Z . Then lq,ξ,E −m, x, χ Em,q,ξ,χ x.
Proof (cf. [17, 18]). Let χ be a Dirichlet character with odd conductor d and let F be an odd
number of multiple d. Set s ∈ C and x ∈ R . Beside the multiple twisted q-Euler lh
h
function lq,ξ,E s, x, χ, we consider the multiple twisted q-Euler zeta function ζq,ξ,E s, x in
Definition 2.1. Then
h
lq,ξ,E s, x, χ
F
−s
1
Fh−q
a1 · · · ah x
a1 ···ah h
χ a1 · · · ah −ξq
ζqF ,ξF ,E s,
F
a1 ,...,ah 0
F−1
2.11
cf. 14.
In the integral for Γs, we make the change of variable y x l1 · · · lh t, where
l1 , . . . , lh ≥ 0, to obtain
∞
dy
Γs e−y ys
y
0
2.12
s ∞ −xl1 ···lh t s dt
,
e
t
x l1 · · · lh
t
0
Min-Soo Kim et al.
7
or
x l1 · · · lh
−s
Γs ∞
dt
.
t
e−xl1 ···lh t ts
0
2.13
Summing over all l1 , . . . , lh ≥ 0, we find
2hq
∞
l1 ···lh −ξq
x l1 · · · lh
−s
Γs l1 ,...,lh 0
2hq
∞
−ξq
l1 ···lh
∞
e−xl1 ···lh t ts
0
l1 ,...,lh 0
dt
.
t
2.14
This gives us
h
ζq,ξ,E s, xΓs
2hq
∞
0
∞
0
∞
−ξq
l1 ···lh
∞
0
l1 ,...,lh 0
ts
2q
ξqe−t
1
dt
t
e−xl1 ···lh t ts
2q
dt
···
e−xt
−t
t
1 ξqe
2.15
h-times
h
ts Fq,ξ −t, x
dt
.
t
If we divide the infinite integral into two parts:
∞
0
h
ts Fq,ξ −t, x
dt
t
1
0
h
ts Fq,ξ −t, x
dt
t
∞
1
h
ts Fq,ξ −t, x
dt
,
t
2.16
it is easily seen that the second term is an entire function on t.
1
h
h
Consider 0 ts Fq,ξ −t, xdt/t. By the definition of En,q,ξ x, we have
h
Fq,ξ −t, x 2
h
∞
tn
q
h
En,q,ξ x−1n e−xt .
−t
n!
1
ξqe
n0
2.17
Therefore,
1
0
1
∞ E
x
dt n,q,ξ
−1n tsn−1 dt
t
n!
0
n0
h
h
ts Fq,ξ −t, x
h
∞ E
x −1n
n,q,ξ
n0
n!
sn
2.18
.
This has an analytic continuation to a meromorphic function s in the entire complex plane.
It is holomorphic except at s 0, −1, −2, . . . , where it has a pole of order 1. Note that Γs
is holomorphic except at s 0, −1, −2, . . . , where it has a pole of order 1. Γs does not have
h
a zero. Therefore, ζq,ξ,E s, x has an analytic continuation to the whole complex plane. For an
integer m ∈ Z , we have
h
h
Em,q,ξ x
−1m .
lim s m ζq,ξ,E s, xΓs s→−m
m!
2.19
8
Abstract and Applied Analysis
If m ∈ Z , we have lims→−m s mΓs −1m 1/m!, and thus we obtain
h
h
ζq,ξ,E −m, x Em,q,ξ x.
2.20
Consequently, by using Propositions 2.3 and 2.6 and the above equation, we have
h
lq,ξ,E −m, x, χ
F
m
1
F−1
h
χ a1 · · · ah −ξqa1 ···ah Em,qF ,ξF
Fh−q a1 ,...,ah 0
a1 · · · ah x
F
h
Em,q,ξ,χ x.
2.21
Therefore, we obtain another proof of Theorem 2.5.
Remark 2.6 see 11, 14. We put
D
d
.
dt
2.22
Let m ∈ Z , and let x ∈ R . From 2.1 and 2.4, we obtain the following:
h
h
Em,q,ξ x Dm Fq,ξ t, x|t0
2hq
∞
m
−ξql1 ···lh x l1 · · · lh
2.23
l1 ,...,lk 0
h
ζq,ξ,E −m, x.
Similarly, by 2.5 and 2.11, we have
h
h
Em,q,ξ,χ x Dm Fq,ξ,χ t, x|t0
2hq
∞
h
m
χ l1 · · · lh
−1li ξli qli x l1 · · · lh
l1 ,...,lh 0
2.24
i1
h
lq,ξ,E −m, x, χ.
3. Partial multiple twisted q-Euler ζ-functions
Let s ∈ C and ai , F ∈ Z with F as an odd integer and 0 < ai < F, where i 1, . . . , h. Then,
partial multiple twisted q-Euler ζ-functions are as follows cf. 14, 16, 18–20:
h Hq,ξ,E s, a1 , . . . , ah , x | F 2hq
∞
l1 ,...,lh 0
li ≡ai mod F,i1,...,h
−ξql1 ···lh
l1 · · · lh x
s .
3.1
Min-Soo Kim et al.
9
h
h
We give a relationship between Hq,ξ,E s, a1 , . . . , ah , x | F and ζqF ,ξF ,E s, x as follows. For i 1, . . . , h, substituting li ai ni F with F as an odd into 3.1, we have
h Hq,ξ,E s, a1 , . . . , ah , x | F
2hq
∞
n1 ,...,nh 0
1
Fh−q
1
Fh−q
−ξqa1 n1 F···ah nh F
a1 n1 F · · · ah nh F x
s
n ···nh
∞
−ξqa1 ···ah
−ξF qF 1
h
2
s
F
q
Fs
n1 ,...,nh 0 n1 · · · nh a1 · · · ah x /F
3.2
−ξqa1 ···ah h
a1 · · · ah x
.
s,
ζ
qF ,ξF ,E
Fs
F
By using 2.3 and Theorem 2.2 and substituting s −m, m ∈ Z in the above, we
arrive at the following theorem.
Theorem 3.1. Let F be an odd integer, s ∈ C and let x ∈ R . Then
a ···ah
1 −ξq 1
a1 · · · ah x
h h
Hq,ξ,E s, a1 , . . . , ah , x | F .
ζ
s,
qF ,ξF ,E
Fs
F
Fh−q
3.3
In particular, if m ∈ Z , then
h Hq,ξ,E − m, a1 , . . . , ah , x | F
k
m m F
m
h
a1 ···ah −ξq
·
·
·
a
x
Ek,qF ,ξF .
a
1
h
h
k
a
·
·
·
a
x
1
h
F−q
k0
1
3.4
By using Theorem 3.1 and 2.11, we arrive at the following theorem.
Theorem 3.2. Let χ be a Dirichlet character with conductor d and F as an odd multiple of d. Then,
h
lq,ξ,E s, x, χ F−1
h χ a1 · · · ah Hq,ξ,E s, x, a1 , . . . , ah , x | F ,
3.5
a1 ,...,ah 0
where s ∈ C and x ∈ R .
4. Multiple twisted p-adic q-Euler l-functions
Let p be an odd prime. Zp , Qp and Cp will always denote, respectively, the ring of p-adic
integers, the field of p-adic numbers, and the completion of the algebraic closure of Qp . Let vp :
Cp → Q ∪ {∞} Q the field of rational numbers denote the p-adic valuation of Cp normalized
so that vp p 1. The absolute value on Cp will be denoted as |·|p , and |x|p p−vp x for x ∈ Cp .
We let Z×p {x ∈ Zp | 1/x ∈ Zp }. A p-adic integer in Z×p is sometimes called a p-adic unit. For
each integer N ≥ 0, CpN will denote the multiplicative group of the primitive pN -th roots of
unity in C×p Cp \ {0}. Set
N
CpN .
4.1
Tp ξ ∈ Cp | ξp 1, for some N ≥ 0 N≥0
10
Abstract and Applied Analysis
The dual of Zp , in the sense of p-adic Pontrjagin duality, is Tp Cp∞ , the direct limit under
inclusion of cyclic groups CpN of order pN N ≥ 0, with the discrete topology.
When one talks of q-extension, q is variously considered as an indeterminate, a
complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ Cp , then we normally assume
|1 − q|p < 1.
h
We will consider the p-adic analogue of the lq,ξ,E -functions which are introduced in the
h
previous section. In order to consider p-adic and complex lq,ξ,E -functions simultaneously, we
will use an isomorphism, σ, between the algebraic closure of the rational numbers in Cp and
the algebraic closure of the rational numbers within the complex numbers C. Our purpose is
h
to discuss the values of lq,ξ,E -functions, so we will consider σ as fixed throughout this section
and use σ to identify p-adic algebraic numbers with complex algebraic numbers. We will
write x y, when x ∈ Cp , y ∈ C and y σx.
Let ω be denoted as the Teichmüller character having conductor p. For an arbitrary
character χ, let χn χω−n , where n ∈ Z, in sense of the product of characters. We put
a ω−1 aa a
,
ωa
4.2
whenever a, p 1. We then have a ≡ 1 mod pZp for these values of a. Note that we
extend this notation by defining
a pt ω−1 aa pt
4.3
for all a ∈ Z with a, p 1, and t ∈ Cp such that |t|p ≤ 1. Thus, a pt a pω−1 at, so
that a pt ≡ 1 mod pZp t cf. 21, 22.
The significance of Theorem 3.1 lies in the fact that the right-hand side is essentially a
liner combination of terms of the form
m m
k
k0
F
a1 · · · ah pt
k
h
Ek,qF ,ξF ,
4.4
which makes sense when m is replaced by a p-adic variable and p|F. Set
D s ∈ Cp | |s|p ≤ pp−2/p−1 ,
4.5
cf. 8, 16, 18–22. Let F be an odd integer, and let a1 · · · ah , p 1, ai ∈ Z with 0 < ai < F
for i 1, . . . , h. Suppose that s ∈ D and t ∈ Cp with |t|p ≤ 1. We apply 18, Proposition 5.8,
page 53 to the series
∞ s
k0
k
F
a1 · · · ah pt
k
h
Ek,qF ,ξF .
4.6
Let q ∈ 1 Mp where Mp {z ∈ Cp | |z| < 1}. In 13, we see that
h E F F ≤ 1,
k,q ,ξ
p
4.7
Min-Soo Kim et al.
11
since ξ ∈ Tp . Observe that we have for odd p|F,
k
k
F
1
h
−k
a · · · a pt Ek,qF ,ξF ≤ p p ,
1
h
p
4.8
so that we can take r 1/p and M 1 in 18, Proposition 5.8. This prove that 4.6 is analytic
in D. Note that a1 · · · ah pt−s is analytic in D for a1 · · · ah , p 1 and t ∈ Cp such
that |t|p ≤ 1.
Definition 4.1. Let F be an odd integer, and let a1 · · · ah , p 1, ai ∈ Z with 0 < ai < F
for i 1, . . . , h. Suppose that s ∈ D and t ∈ Cp with |t|p ≤ 1. One defines the partial multiple
twisted p-adic q-Euler ζ-functions for p | F:
h
Hp,q,ξ,E s, a1 , . . . , ah , pt | F
k
∞ −s −ξqa1 ···ah F
−s
h
·
·
·
a
pt
Ek,qF ,ξF .
a
1
h
h
k
a
·
·
·
a
pt
1
h
F−q
k0
4.9
Theorem 4.2. Let F be an odd integer with p | F, and let a1 · · ·ah , p 1, ai ∈ Z with 0 < ai < F
h
for i 1, . . . , h. Suppose that s ∈ D and t ∈ Cp with |t|p ≤ 1. Then Hp,q,ξ,E s, a1 , . . . , ah , pt | F is a
p-adic analytic function on D such that
h h
Hp,q,ξ,E − m, a1 , . . . , ah , pt | F ω−m a1 · · · ah Hq,ξ,E − m, a1 , . . . , ah , pt | F
4.10
for m ∈ Z . In particular, if m ≡ 0 mod p − 1, then
h
h Hp,q,ξ,E − m, a1 , . . . , ah , pt | F Hq,ξ,E − m, a1 , . . . , ah , pt | F .
4.11
Proof. We have already remarked that a1 · · · ah pt−s is analytic in D for a1 · · ·ah , p h
1 and t ∈ Cp such that |t|p ≤ 1. Also we see that 4.6 is analytic in D. It is clear that Hp,q,ξ,E is
a p-adic analytic function on D. For s −m, m ∈ Z one has
h
Hp,q,ξ,E − m, a1 , . . . , ah , pt | F
−ξqa1 ···ah
Fh−q
a1 · · · ah pt
ωa1 · · · ah m k
m F
m
h
Ek,qF ,ξF
k
a
·
·
·
a
pt
1
h
k0
where we use Theorem 3.1
h ω−m a1 · · · ah Hq,ξ,E − m, a1 , . . . , ah , pt | F .
This completes the proof.
4.12
12
Abstract and Applied Analysis
Definition 4.3. Let χ be a Dirichlet character with odd conductor d, and let F be a positive
multiple of p and d. We can define the multiple twisted p-adic q-Euler l-function:
F−1
h
lp,q,ξ,E s, t, χ h χ a1 · · · ah Hp,q,ξ,E s, a1 , . . . , ah , pt | F .
a1 ,...,ah 0
a1 ···ah ,p1
4.13
Theorem 4.4. Let χ be a Dirichlet’s character with an odd conductor d, and let F be a positive multiple
h
of p and d. Then lq,ξ,E s, t, χ is a p-adic analytic function on D with
h
h
lp,q,ξ,E −m, t, χ Em,q,ξ,χm pt −
where m ∈ Z and
I0 Fm
Fh−q
χm p
β∈I0
β h
χm β − ξp qp Em,qF ,ξF
βt
,
F/p
x a1 · · · ah ≡ 0 mod p
1 x
,
p for some a1 , . . . , ah with 0 ≤ a1 , . . . , ah ≤ F − 1
4.14
4.15
and in β∈I0 one sums over β 1/px as many times as x is expressed in the form x a1 · · · ah
by various aj ’s, and χm χω−m with ω the Teichmüller in the sense of the product of characters.
Remark 4.5. Theorem 4.4 can be extended to obtain similar results for the multiple p-adic Lfunction in 16. In the case h 1, we note that
β∈I0
F/p−1
4.16
.
a0
Observe that if h 1, then
h
lim lp,q,ξ,E −m, t, χ
h→1
Em,q,ξ,χm pt −
F/p−1
a
Fm
at
χm p
χm a − ξp qp Em,qF ,ξF
F−q
F/p
a0
Em,q,ξ,χm pt − pm χm p
F/p−qp F m
F−q
p
F/p−1
a
1
at
χm a − ξp qp Em,qp F/p ,ξp F/p
F/p−qp a0
F/p
where we use Proposition 2.3
Em,q,ξ,χm pt − pm
1
χm pEm,qp ,ξp ,χm t.
p−q
4.17
This function interpolates the twisted generalized q-Euler polynomials at negative integers.
1
For lp,q,ξ,E −m, t, χ, the twisted p-adic q-Euler l-functions similar results were obtained cf.
see for detail 8, Theorem 9. If q → 1 in the above, then
1
lim lp,q,ξ,E s, t, χ lp,ξ,E s, t, χ 2
q→1
∞
χl−1l ξl
,
t ls
l0
l,p1
which is called the twisted p-adic lE -function of two variables.
|t|p ≤ 1,
4.18
Min-Soo Kim et al.
13
h
Proof. The formula for lq,ξ,E s, x, χ is p-adic analytic function in D by the Theorem 4.2. On the
other hand, by substituting s −m, m ∈ Z , into Definition 4.3, we have
F−1
h
lp,q,ξ,E −m, t, χ h χ a1 · · · ah Hp,q,ξ,E − m, a1 , . . . , ah , pt | F
a1 ,...,ah 0
a1 ···ah ,p1
F−1
h χ a1 · · · ah Hp,q,ξ,E − m, a1 , . . . , ah , pt | F
4.19
a1 ,...,ah 0
F−1
−
a1 ,...,ah 0
1
a1 ···ah ,p /
h χ a1 · · · ah Hp,q,ξ,E − m, a1 , . . . , ah , pt | F .
From Theorems 2.2, 3.1, and 4.2, we obtain
F−1
a1 ,...,ah 0
1
a1 ···ah ,p /
h χ a1 · · · ah Hp,q,ξ,E − m, a1 , . . . , ah , pt | F
F−1
a1 ,...,ah 0
1
a1 ···ah ,p /
Fm
Fh−q
F m h
χm a1 · · · ah −ξqa1 ···ah
Em,qF ,ξF
Fh−q
χm p
p p β
χm β − ξ q
β∈I0
h
Em,qF ,ξF
a1 · · · ah pt
F
4.20
βt
.
F/p
Therefore,
h
lp,q,ξ,E −m, t, χ
h
Em,q,ξ,χm pt
−
Fm
Fh−q
χm p
β∈I0
p p β
χm β − ξ q
h
Em,qF ,ξF
βt
.
F/p
4.21
This completes the proof.
Acknowledgment
This work is supported by Kyungnam University Foundation Grant, 2008.
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