Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 358986, 13 pages
doi:10.1155/2010/358986
Research Article
On Carlitz’s Type q-Euler Numbers Associated with
the Fermionic P -Adic Integral on Zp
Min-Soo Kim,1 Taekyun Kim,2 and Cheon-Seoung Ryoo3
1
Department of Mathematics, Korea Advanced Institute of Science and Technology,
373-1 Guseong-dong, Yuseong-Gu, Daejeon 305-701, Republic of Korea
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 2 August 2010; Accepted 28 September 2010
Academic Editor: Jewgeni Dshalalow
Copyright q 2010 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.
We consider the following problem in the paper of Kim et al. 2010: “Find Witt’s formula for
Carlitz’s type q-Euler numbers.” We give Witt’s formula for Carlitz’s type q-Euler numbers, which
is an answer to the above problem. Moreover, we obtain a new p-adic q-l-function lp,q s, χ for
Dirichlet’s character χ, with the property that lp,q −n, χ En,χn ,q − χn ppnq En,χn ,qp for n 0, 1, . . .
using the fermionic p-adic integral on Zp .
1. Introduction
Throughout this paper, let p be an odd prime number. The symbol, Zp , Qp , and Cp denote the
rings of p-adic integers, the field of p-adic numbers, and the field of p-adic completion of the
algebraic closure of Qp , respectively. The p-adic absolute value in Cp is normalized in such
way that |p|p p−1 . Let N be the set of natural numbers and Z N ∪ {0}.
As the definition of q-number, we use the following notations:
1 − qx
,
xq 1−q
x−q
x
1 − −q
.
1q
1.1
Note that limq → 1 xq x for x ∈ Zp , where q tends to 1 in the region 0 < |q − 1|p < 1.
When one talks of q-analogue, q is variously considered as an indeterminate,
a complex number q ∈ C, or a p-adic number q ∈ Cp . If q 1 t ∈ Cp , one normally assumes
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Journal of Inequalities and Applications
|t|p < 1. We will further suppose that ordp t > 1/p − 1, so that qx expx log q for |x|p ≤ 1.
If q ∈ C, then we assume that |q| < 1.
After Carlitz 1, 2 gave q-extensions of the classical Bernoulli numbers and
polynomials, the q-extensions of Bernoulli and Euler numbers and polynomials have been
studied by several authors cf. 1–21. The Euler numbers and polynomials have been
studied by researchers in the field of number theory, mathematical physics, and so on cf.
1, 2, 9, 11, 13–16, 22, 23. Recently, various q-extensions of these numbers and polynomials
have been studied by many mathematicians cf. 6–8, 10, 12, 17, 18, 20. Also, some authors
have studied in the several area of q-theory cf. 3, 4, 16, 19, 24.
It is known that the generating function of Euler numbers Ft is given by
Ft ∞
tn
2
En .
t
e 1 n0 n!
1.2
From 1.2, we know the recurrence formula of Euler numbers is given by
E 1n En 0 if n > 0,
E0 1,
1.3
with the usual convention of replacing En by En see 7, 18.
∗
are defined as
In 17, the q-extension of Euler numbers En,q
∗
E0,q
1,
⎧
⎨2 if n 0,
n
∗
qE∗ 1 En,q
⎩0 if n > 0,
1.4
∗
with the usual convention of replacing E∗ n by En,q
.
As the same motivation of the construction in 18, Carlitz’s type q-Euler numbers En,q
are defined as
E0,q
2
,
2q
⎧
⎨2 if n 0,
n
q qE 1 En,q ⎩0 if n > 0,
1.5
with the usual convention of replacing En by En,q . It was shown that limq → 1 En,q En , where
En is the nth Euler number. In the complex case, the generating function of Carlitz’s type
q-Euler numbers Fq t is given by
Fq t ∞
∞
n n t
tn
En,q 2
−q e q ,
n!
n0
n0
1.6
where q is a complex number with |q| < 1 see 18. The remark point is that the series on
the right-hand side of 1.6 is uniformly convergent in the wider sense. In p-adic case, Kim
et al. 18 could not determine the generating function of Carlitz’s type q-Euler numbers and
Witt’s formula for Carlitz’s type q-Euler numbers.
In this paper, we obtain the generating function of Carlitz’s type q-Euler numbers
in the p-adic case. Also, we give Witt’s formula for Carlitz’s type q-Euler numbers, which
Journal of Inequalities and Applications
3
is a partial answer to the problem in 18. Moreover, we obtain a new p-adic q-l-function
lp,q s, χ for Dirichlet’s character χ, with the property that
n
lp,q −n, χ En,χn ,q − χn p p q En,χn ,qp,
1.7
for n ∈ Z using the fermionic p-adic integral on Zp .
2. Carlitz’s Type q-Euler Numbers in the p-Adic Case
Let UDZp be the space of uniformly differentiable functions on Zp . Then, the p-adic qintegral of a function f ∈ UDZp on Zp is defined by
Iq f Zp
fadμq a lim N →∞
1
pN
N
−1
p
faqa ,
2.1
q a0
cf. 5–17, 19, 20, 22. The bosonic p-adic integral on Zp is considered as the limit q → 1, that
is,
I1 f Zp
fadμ1 a.
2.2
From 2.1, we have the fermionic p-adic integral on Zp as follows:
I−1 f lim Iq f q → −1
Zp
fadμ−1 a.
2.3
Using 2.3, we can readily derive the classical Euler polynomials, En x, namely
2
∞
tn
2ext
exyt dμ−1 y t
En x .
n!
e 1 n0
Zp
2.4
In particular, when x 0, En 0 En is the well-known the Euler numbers cf. 7, 16, 19.
By definition of I−1 f, we show that
I−1 f1 I−1 f 2f0,
2.5
where f1 x fx 1 see 7. By 2.5 and induction, we obtain
n−1
I−1 fn −1n−1 I−1 f 2 −1n−i−1 fi,
i0
2.6
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Journal of Inequalities and Applications
where n 1, 2, . . . and fn x fx n. From 2.6, we note that
n−1
I−1 fn I−1 f 2 −1i fi
if n is odd
i0
n−1
I−1 fn − I−1 f 2 −1i1 fi
2.7
if n is even.
i0
For x ∈ Zp and any integer i ≥ 0, we define
⎧
⎪
⎨ xx − 1 · · · x − i 1
i!
⎪
⎩1,
x
i
if i ≥ 1,
if i 0.
2.8
It is easy to see that xi ∈ Zp see 23, page 172. We put x ∈ Cp with ordp x > 1/p − 1
and |1 − q|p < 1. We define qx for x ∈ Zp by
q x
∞
x i
q−1 ,
i
i0
xq ∞
x i1
i
i−1
.
q−1
2.9
If we set fx qx in 2.7, we have
I−1 qx n−1
2
2 −1i qi n
q 1 i0
q1
I−1 qx n−1
2
2 −1i1 qi qn − 1 i0
q1
if n is odd
2.10
if n is even.
From 2.10, we note that if fx qx , then I−1 qx 2/q 1, hence there is no need to
consider both odd and even cases. Thus, for each l ∈ N, we obtain I−1 qlx 2/ql 1.
Therefore, we have
I−1 q
x
xnq
1
1−q
1
1−q
n
n
n
−1l I−1 ql1x
l
n
n
l0
n
l0
l
2.11
2
.
−1l l1
q 1
Also, if fx qlx in 2.5, then
I−1 qlx1 I−1 qlx 2f0 2.
2.12
Journal of Inequalities and Applications
5
On the other hand, by 2.12, we obtain that
I−1 q
x1
x 1nq
I−1 q
x
xnq
1
1−q
2
1−q
n
n
n
−1
l
l0
n
l
I−1
q
l1
x1 I−1
q
l1
x n
n
−1l 0
l
l0
2.13
is equivalent to
0 I−1 qx1 x 1nq I−1 qx xnq
qI−1 qx 1 qxn I−1 qx xnq
qI−1 q
x
n
n
l
l0
q
q x
l
l
I−1 qx xnq
2.14
n
n
ql I−1 qx xl I−1 qx xnq .
l
l0
From the definition of fermionic p-adic integral on Zp and 2.11, we can derive
I−1 q
x
xnq
Zp
xnq qx dμ−1 x
lim
N
−1
p
N →∞
a0
1
1−q
1
1−q
n
1
1−q
i
i0
a
−1i qia −q
n
n
i0
n
n
n
n
−1i lim
N →∞
i
n
n
i0
i
−1i
N
−1
p
−1a qi1
a
2.15
a0
2
1 qi1
is equivalent to
∞
∞
n
n n
1
2
tn
n t
x
I−1 q xq
−1i
n
n! n0 1 − q i0 i
1 qi1 n!
n0
∞
n n t
2
−q e q .
n0
2.16
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Journal of Inequalities and Applications
From 2.12, 2.13, 2.14, 2.15, and 2.16, it is easy to show that
q
n
n
l0
l
ql El,q En,q ⎧
⎨2
if n 0,
⎩0
if n > 0,
2.17
where En,q are Carlitz’s type q-Euler numbers defined by see 18
∞
∞
n n t tn
En,q .
Fq t 2
−q e q n!
n0
n0
2.18
Therefore, we obtain the recurrence formula for the Carlitz’s type q-Euler numbers as follows:
n
q qE 1 En,q ⎧
⎨2
if n 0,
⎩0
if n > 0,
2.19
with the usual convention of replacing En by En,q . Therefore, by 2.16, 2.18, and 2.19, we
obtain the following theorem, which is a partial answer to the problem in 18.
Theorem 2.1 Witt’s formula for En,q . For n ∈ Z ,
En,q 1
1−q
n
n
n
i0
i
−1i
2
1 qi1
Zp
xnq qx dμ−1 x.
2.20
Carlitz’s type q-Euler numbers En En,q can be determined inductively by
⎧
⎨2
n
q qE 1 En,q ⎩0
if n 0,
if n > 0,
2.21
with the usual convention of replacing En by En,q .
Carlitz type q-Euler polynomials En,q x are defined by means of the generating
function Fq x, t as follows:
∞
∞
tn
En,q x .
Fq x, t 2 −1k qk ekxq t n!
n0
k0
2.22
In the cases x 0, En,q 0 En,q will be called Carlitz type q-Euler numbers cf. 8, 19. One
also can see that the generating functions Fq x, t are determined as solutions of
Fq x, t 2exq t − qet Fq x, qt .
From 2.22, one gets the following.
2.23
Journal of Inequalities and Applications
7
j xj
j1
tj /j!.
Lemma 2.2. 1Fq x, t 2et/1−q ∞
j0 1/q − 1 q 1/1 q
∞
2En,q x 2 k0 −1k qk k xnq .
It is clear from 1 and 2 of Lemma 2.2 that
n
n −1k xk
En,q x q ,
n
1 − q k0 k 1 qk1
2
m−1
∞
∞
−1k qk k xnq − −1km qkm k m xnq
k0
k0
−1k qk k xnq 2.24
k0
En,q x −1m1 qm En,q x m .
1
2
From 2.24, we may state the following.
Proposition 2.3. If m ∈ N and n ∈ Z , then
(1) En,q x 2/1 − qn nk0 nk −1k /1 qk1 qxk ,
k k
n
m1 m
(2) m−1
q En,q x m.
k0 −1 q k xq 1/2En,q x −1
Proposition 2.4. For n ∈ Z , the value of Zp x ynq qy dμ−1 y is n! times the coefficient of tn in the
k k kxq t
formal expansion of 2 ∞
in powers of t. That is, En,q x Zp x ynq qy dμ−1 y.
k0 −1 q e
Proof. From 2.3, we have
Zp
q
N
−1
p
a
xk
−qk1 q dμ−1 y q lim
kxy y
N →∞
a0
2qxk
,
1 qk1
2.25
which leads to
n
n
n y
1
k
qkxy qy dμ−1 y
x y q q dμ−1 y 2
n −1
1−q
Zp
Zp
k0 k
n
n −1k xk
2
q .
n
1 − q k0 k 1 qk1
2.26
The result now follows by using 1 of Proposition 2.3.
Corollary 2.5. If n ∈ Z , then
En,q x n
n
k0
k
kx
xn−k
q q Ek,q .
2.27
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Journal of Inequalities and Applications
Let d ∈ N with d ≡ 1 mod 2 and p be a fixed odd prime number. One sets
X lim
←
N
Z
,
dpN Z
X∗ a dpZp ,
0<a<dp
a,p1
2.28
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z with 0 ≤ a < dpN cf. 7, 9. Note that the natural map Z/dpN Z → Z/pN Z
induces
π : X −→ Zp .
2.29
Hereafter, if f is a function on Zp , one denotes by the same f the function f ◦ π on X. Namely
one considers f as a function on X.
Let χ be the Dirichlet character with an odd conductor d dχ ∈ N. Then, the
generalized Carlitz type q-Euler polynomials attached to χ are defined by
En,χ,q x X
n
χa x y q qy dμ−1 y ,
2.30
where n ∈ Z and x ∈ Zp . Then, one has the generating function of generalized Carlitz type
q-Euler polynomials attached to χ
Fq,χ x, t 2
∞
χm−1m qm emxq t m0
∞
tn
En,χ,q x .
n!
n0
2.31
Now, fixed any t ∈ Cp with ordp t > 1/p − 1 and |1 − q|p < 1. From 2.31, one has
Fq,χ x, t 2
∞
χm −q
∞
m 1
n
n
n
1 − q i0
∞
n
n
1
2 −1i qix
n
1
−
q
i
n0
i0
m0
n0
i
−1i qimx
tn
n!
d−1 ∞
jdl ijdl tn
×
χ j dl −q
q
n!
j0 l0
n
d−1
n
j qixj tn
,
−1i
n χ j −q
1 qdi1 n!
1 − q j0
i
i0
∞
2 n0
1
2.32
Journal of Inequalities and Applications
9
where x ∈ Zp and d ∈ N with d ≡ 1 mod 2. By 2.31 and 2.32, one can derive
n
d−1
n
j 2
En,χ,q x −1i qixj
n χ j −q
1 qdi1
1 − q j0
i
i0
1
N
p
−1
n
d−1
l
n
j i ixj
χ
j
−q
q
×
lim
−1l qdi1
−1
n
N →∞
1 − q j0
i
i0
l0
1
−1
d−1 p
lim
χ j dl N
N →∞
lim
N →∞
X
1
1−q
j0 l0
N
dp
−1
1
χa 1−q
a0
n
n
n
n
i0
i
−1i qijdlx × −1jdl qjdl
2.33
n
n
i0
i
a
−1i qiax −q
n
χ y x y q qy dμ−1 y ,
where x ∈ Zp and d ∈ N with d ≡ 1 mod 2. Therefore, one obtains the following.
Theorem 2.6.
n
d−1
n
j 2
,
En,χ,q x −1i qixj
n χ j −q
di1
1
q
1 − q j0
i
i0
1
2.34
where n ∈ Z and x ∈ Zp .
Let ω denote the Teichmüller character mod p. For x ∈ X ∗ , one sets
x xq ω−1 x xq
ωx
.
2.35
Note that since |x − 1|p < p−1/p−1 , xs is defined by expslogp x for |s|p ≤ 1 cf. 10, 12,
21. One notes that xs is analytic for s ∈ Zp .
One defines an interpolation function for Carlitz type q-Euler numbers. For s ∈ Zp ,
lp,q s, χ X∗
x−s χxqx dμ−1 x.
Then, lp,q s, χ is analytic for s ∈ Zp .
The values of this function at nonpositive integers are given by the following.
2.36
10
Journal of Inequalities and Applications
Theorem 2.7. For integers n ≥ 0,
n
lp,q −n, χ En,χn ,q − χn p p q En,χn ,qp ,
2.37
where χn χω−n . In particular, if χ ωn , then lp,q −n, ωn En,q − pnq En,qp .
Proof.
lp,q
−n, χ X∗
X
X
xn χxqx dμ−1 x
n px
px q χn px q dμ−1 px
xnq χn xqx dμ−1 x
−
xnq χn xqx dμ−1 x
n − p q χn p
X
2.38
X
xnqp χn xqpx dμ−1 x.
Therefore by 2.30, the theorem is proved.
Let χ be the Dirichlet character with an odd conductor d dχ ∈ N. Let F be a positive
integer multiple of p and d. Then, by 2.22 and 2.31, we have
Fq,χ x, t 2
∞
χm−1m qm emxq t
m0
∞
F−1
a Fk Fq kxa/F F t
q
2 χa −q
−q e
a0
∞
k0
Fnq
n0
F−1
a
χa −q En,qF
x a
a0
F
2.39
tn
.
n!
Therefore, we obtain the following
En,χ,q x Fnq
x a
a
.
χa −q En,qF
F
a0
F−1
2.40
If χn p / 0, then p, dχn 1, so that F/p is a multiple of dχn . From 2.40, we derive
n
n F n F/p−1
a
a
χn a −qp En,qp F/p
χn p p q En,χn ,qp χn p p q
p qp a0
F/p
Fnq
F
a
a
.
χn a −q En,qF
F
a0
p|a
2.41
Journal of Inequalities and Applications
11
Thus, we have
F−1
a
a
n
.
En,χn ,q − χn p p q En,χn ,qp Fnq χn a −q En,qF
F
a0
2.42
pa
By Corollary 2.5, we easily see that
En,qF
a
F
n
n a n−k
k0
F
k
n
F−n
q aq
qF
qka Ek,qF
k
n
n
F
k0
k
a
qa
2.43
ka
q Ek,qF .
From 2.42 and 2.43, we have
F−1
a
a
n
En,χn ,q − χn p p q En,χn ,qp Fnq χn a −q En,qF
F
a0
pa
k
∞
n
a F
n
χaa −q
qka Ek,qF ,
a qa
a0
k0 k
F−1
2.44
pa
since χn a χaω−n a. From Theorem 2.7 and 2.44, we have
lp,q
k
∞
F−1
n
a F
n
χaa −q
qka Ek,qF ,
−n, χ a
a
q
a0
k0 k
2.45
pa
for n ∈ Z . Therefore, we have the following theorem.
Theorem 2.8. Let F be a positive integer multiple of p and d dχ , and let
lp,q
s, χ X∗
x−s χxqx dμ−1 x,
s ∈ Zp .
2.46
Then, lp,q s, χ is analytic for s ∈ Zp and
lp,q
k
∞
F−1
−s
a F
−s χaa −q
qka Ek,qF .
s, χ a
a
k
q
a0
k0
2.47
pa
Furthermore, for n ∈ Z
n
lp,q −n, χ En,χn ,q − χn p p q En,χn ,qp .
2.48
12
Journal of Inequalities and Applications
Acknowledgments
The first author was supported by the Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science, and
Technology 2010-0001654. The second author was supported by the research grant of
Kwangwoon University in 2010.
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