Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 581582, 11 pages
doi:10.1155/2008/581582
Research Article
Euler Numbers and Polynomials Associated
with Zeta Functions
Taekyun Kim
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Correspondence should be addressed to Taekyun Kim, tkim@knu.ac.kr
Received 16 January 2008; Accepted 19 April 2008
Recommended by Lance Littlejohn
zeta function and the
Euler zeta function are defined by
For s ∈ C
, the Euler
Hurwitz-type
n
n
s
∞
s
ζE s 2 ∞
n1 −1 /n , and ζE s, x 2
n0 −1 /n x . Thus, we note that the Euler
zeta functions are entire functions in whole complex s-plane, and these zeta functions have the
values of the Euler numbers or the Euler polynomials at negative integers. That is, ζE −k Ek∗ ,
and ζE −k, x Ek∗ x. We give some interesting identities between the Euler numbers and the zeta
functions. Finally, we will give the new values of the Euler zeta function at positive even integers.
Copyright q 2008 Taekyun Kim. 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
Throughout this paper, Z, Q, C, Zp , Qp , and Cp will, respectively, denote the ring of rational
integers, the field of rational numbers, the field of complex numbers, the ring p-adic rational
integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp .
Let vp be the normalized exponential valuation of Cp such that |p|p p−vp p p−1 . If q ∈ Cp ,
we normally assume that |q − 1|p < 1. We use the notation
1 − qx
1 − −qx
,
x−q .
1.1
1−q
1q
Hence, limq→1 xq 1, for any x with |x|p ≤ 1 in the present p-adic case.
Let p be a fixed odd prime. For d odd, a fixed positive integer with p, d 1, let
Z
X Xd lim
,
X1 Zp ,
X∗ a dpZp
←
N
0<a<dp
N dp Z
1.2
a,p1
N
n
a dp Zp x ∈ X | x ≡ a mod dp ,
xq where a ∈ Z lies in 0 ≤ a < dpN .
2
Abstract and Applied Analysis
In 1, we note that
−qa
−1a qa
μ−q a dpN Zp 1 q
N
dpN −q
1 qdp
1.3
is distribution on X for q ∈ Cp with |1 − q|p < 1. This distribution yields an integral as follows:
−1
p
1
I−q f fxdμ−q x lim N fx−qx ,
N→∞ p
Zp
−q x0
N
for f ∈ UD Zp ,
1.4
which has a sense as we see readily that the limit is convergent see 1. Let q 1. Then, we
have the fermionic p-adic integral on Zp as follows:
I−1 f Zp
fxdμ−1 x lim
N
−1
p
N→∞
fx−1x ,
1.5
x0
cf. 1–5. For any positive integer N, we set
qa
μq a lpN Zp N lp q
1.6
cf. 1–3, 6–20 and this can be extended to a distribution on X. This distribution yields p-adic
bosonic q-integral as follows see 11, 20:
Iq f fxdμq x fxdμq x,
1.7
Zp
X
where f ∈ UDZp the space of uniformly differentiable function on Zp with values in Cp ,
cf. 2, 11, 16–20. In view of notation, I−1 can be written symbolically as I−1 f limq→−1 Iq f.
If we take fx q−x xnq , then we can derive the q-extension of Bernoulli numbers and
polynomials from p-adic q-integrals on Zp as follows:
βn,q Zp
q−x xnq dμq x,
βn,q x Zp
q−y y xq dμq y
1.8
i
m
,
i iq
1.9
cf. 11, 20. Thus, we note that
β0,q
q−1
,
log q
βm,q
m
1
m
q − 1 i0
cf. 11, 14, 20. In the complex plane, the ordinary Bernoulli numbers are a sequence of signed
rational numbers that can be defined by the identity
∞
t
tn
B
,
n
et − 1 n0 n!
cf. 1–33.
|t| < 2π
1.10
Taekyun Kim
3
These numbers arise in the series expansions of trigonometric functions, and are
extremely important in number theory and analysis. From the generating function of Bernoulli
numbers, we note that B0 1, B1 −1/2, B2 1/6, B4 −1/30, B6 1/42, B8 −1/30, B10 5/66, B12 −691/2730, B14 7/6, B16 −3617/510, B18 43867/798, B20 −174611/330, . . . ,
and B2k1 0 for k ∈ N. It is well known that Riemann zeta function is defined by
ζs ∞
1
,
s
n
n1
for s ∈ C.
1.11
We also note that the Riemann zeta function is closely related to Bernoulli numbers at positive
integer or negative integer in the complex plane. Riemann did develop the theory of analytic
continuation needed to rigorously define ζs for all s ∈ C − {0}. From this zeta function, he
derived the following formula cf. 1–33:
ζ−n −
Bn1
,
n1
n ∈ N {1, 2, 3, . . .}.
1.12
Thus, we note that ζ−n 0 if n is an even integer and greater than 0. These are called the
trivial zeros of the zeta function. In 1859, starting with Euler’s factorization of the zeta function
ζs 1
,
1
−
p −s
p:prime
1.13
he derived an explicit formula for the prime numbers in terms of zeros of the zeta function. He
also posed the Riemann hypothesis: if ζz 0, then either z is a trivial zero or z lies on the
critical line Rez 1/2 cf. 4, 5, 16–20, 27–33. It is well known that
sin z
z
z2
z3
z2
1−
1
−
....
1− 2
π
2π2
3π2
1.14
Thus,
1 − z cot z 2
∞
ζ2m
m1
π 2m
1.15
z2m
cf. 4, 5, 10, 16–20, 27–33. From this, we can derive the following famous formula.
Lemma 1.1. For n ∈ N,
ζ2n ∞
1
−1n−1 2π2n
B2n ,
22n!
k 2n
k1
for n ∈ N.
1.16
It is easy to see that
z cot z ∞
2iz
−1k 22k B2k 2k
iz
1
z .
2k!
e2iz − 1
k1
1.17
4
Abstract and Applied Analysis
However, it is not known the values of ζ2k 1 for k ∈ N. In the case of k 1, Apery proved
that ζ3 is irrational number see 34. The constants Ek∗ in the Taylor series expansion
et
∞
tn
2
En∗ ,
1 n0 n!
where |t| < π
1.18
cf. 3–5, 10, 27 are known as the first-kind Euler numbers. From the generating function of
the first-kind Euler numbers, we note that
n
n
∗
∗
E0 1, En −
1.19
El∗ , for n ∈ N.
l
l0
∗
The first few are 1, − 1/2, 0, 1/4, . . . , and E2k
0 for k 1, 2, . . . . The Euler polynomials are
also defined by
∞
∞
n
2
tn tn
n
xt
∗
∗ n−k
.
1.20
e En x Ek x
t
n! n0 k0 k
n!
e 1
n0
For s ∈ C, the Euler zeta function and Hurwitz’s type Euler zeta function are defined by
ζE s 2
∞
−1n
n1
ns
ζE s, x 2
,
∞
−1n
s
n0 n x
1.21
cf. 2, 4, 5, 9, 10, 27. Thus, we note that Euler zeta functions are entire functions in the
whole complex s-plane and these zeta functions have the values of the Euler numbers or Euler
polynomials at negative integers. That is,
ζE −k Ek∗ ,
ζE −k, x Ek∗ x
1.22
cf. 2, 4, 5, 9, 10, 27.
In this paper, we give some interesting identities between Euler numbers and zeta
functions. Finally, we will give the values of the Euler zeta function at positive even integers.
2. Preliminaries/Euler numbers associated with p-adic fermionic integrals
Let f1 x be the translation defined by f1 x fx 1. Then we have
I−1 f1 −I−1 f 2f0.
2.1
If we take fx exyt , then we can derive the first-kind Euler polynomials from the integral
equation of I−1 f as follows:
Zp
exyt dμ−1 y ext
et
∞
En∗ xtn
2
.
1 n0 n!
2.2
That is,
Zp
y n dμ−1 y En∗ ,
Zp
x yn dμ−1 y En∗ x.
2.3
Taekyun Kim
5
For n ∈ N, we have the following integral equation:
Zp
fx ndμ−1 x −1n−1
Zp
fxdμ−1 x 2
n−1
−1n−1l fl.
2.4
l0
From this we note that
Ek∗ n − Ek∗ 2
n−1
−1l−1 lk ,
if n ≡ 0 mod 2,
l0
Ek∗ n
Ek∗
2
n−1
2.5
l k
−1 l ,
if n ≡ 1 mod 2.
l0
Let fx sin ax or fx cos ax. By using the fermionic p-adic q-integral on Zp , we
see that
0
sin ax dμ−1 x sin ax dμ−1 x
Zp
Zp
cos a 1 sin ax dμ−1 x sin a cos ax dμ−1 x,
Zp
2.6
Zp
see 12,
2 cos a 1 cos ax dμ−1 x − sin a sin ax dμ−1 x.
Zp
Zp
2.7
Thus, we obtain
Zp
cos ax dμ−1 x 1,
Zp
sin ax dμ−1 x −
sin a
,
cos a 1
2.8
∗
.
E2n1
2.9
see 12. From this we note that
tan
a
−
2
Zp
sin ax dμ−1 x ∞
−1n1 a2n1
n0
2n 1!
By the same motivation, we can also observe that
a
a
cot 2
2
Zp
cos ax dμ1 x ∞
−1n B2n
n0
2n!
a2n ,
2.10
see 12. These formulae are also treated in Section 3.
Let fx et2x1 . Then we can derive the generating function of the second-kind Euler
numbers from fermionic p-adic integral equation as follows:
Zp
et2x1 dμ−1 x et
∞
tn
2
1
En .
−t
cosh t n0 n!
e
2.11
6
Abstract and Applied Analysis
Thus, we have
E 1n E − 1n 2δ0,n ,
2.12
where we have used the symbolic notation En for En . The first few are E0 1, E1 0, E2 −1, E3 0, E4 5, . . ., E2k1 0 for k ∈ N. In particular,
E2n −
n−1
k0
2n
E2k .
2k
2.13
Recently, Simsek, Ozden, Cangül, Cenkci, Kurt, and others have studied the various extensions
of the first kind Euler numbers by using fernionic p-adic invariant q-integral on Zp , see 2–
5, 16, 20, 27. It seems to be also interesting to study the q-extensions of the second-kind Euler
numbers due to Simsek et al. see 4, 5, 16.
3. Some relationships between Euler numbers and zeta functions
In this section, we also consider Bernoulli and the second Euler numbers in the complex plane.
The second-kind Euler numbers Ek are defined by the following expansion:
sech x ∞
xk
1
2ex
Ek ,
2x
cosh x e 1 k0 k!
for |x| <
π
2
3.1
cf. 10. From 1.18 and 3.1, we can derive the following equation:
k
Ek 2l El∗ ,
l
l0
k
k
where
is binomial coefficient.
l
3.2
By 3.2 and 1.18, we easily see that E0 1, E1 0, E2 −1, E3 0, E4 5, E6 61, . . . , and
E2k1 0 for k 1, 2, 3, . . . . As Euler formula, it is well known that
eix cos x i sin x,
where i −11/2 .
3.3
From 3.3, we note that cos x eix e−ix /2. Thus, we have
sec x ∞ n
i En n
2
sechix
x
ix
−ix
n!
e e
n0
∞
−1n E2n
n0
2n!
x
2n
∞
∞
−1n E2n1 2n1 −1n E2n 2n
i
x
x .
2n 1!
2n!
n0
n0
3.4
From 3.4, we derive
x sec x ∞
−1n E2n
n0
2n!
x2n1 ,
for |x| <
π
.
2
3.5
Taekyun Kim
7
The Fourier series of an odd function on the interval −p, p is the sine series:
∞
fx nπx
,
p
3.6
nπx
dx.
p
3.7
bn sin
n1
where
2
bn p
p
fx sin
0
Let us consider fx sin ax on −π, π. From 3.6 and 3.7, we note that
sin ax ∞
3.8
bn sin nx,
n1
where
π
π
cosn − ax − cosn ax
dx
2
0
0
1 sinn − ax sinn ax π
n
n−1 2
.
−1
−
sin aπ
π
n−a
na
π
n2 − a2
0
bn 2
π
2
π
sin ax sin nx dx 3.9
In 3.8, if we take x π/2, then we have
sin
∞
∞
πa 2n − 1
2
b2n−1 −1n−1 sin aπ −1n−1
2
π
2n
− 12 − a2
n1
n1
∞
∞
∞
2
2n − 1−1n−1
−1n−1 a2k
2
sin aπ
sin
aπ
2
2
π
π
2n − 1 k0 2n − 12k
n1 2n − 1 1 − a/2n − 1
n1
∞
∞
−1n−1
2
sin aπ
a2k
2k1
π
2n
−
1
k0 n1
3.10
From 3.10, we note that
πa
sec
2
πa
2
∞
∞
2
k0
−1n−1
2k1
n1 2n − 1
a2k1 .
3.11
In 3.5, it is easy to see that
πa
sec
2
πa
2
∞
−1n E2n π 2n1
n0
2n!
2
a2n1 .
3.12
By 3.11 and 3.12, we obtain the following.
Theorem 3.1. For n ∈ N,
∞
−1k−1
k1 2k
− 12n1
∞
−1k
k0 2k
12n1
−1n
2n1
π
E2n
.
22n! 2
3.13
8
Abstract and Applied Analysis
It is easy to see that
∞
−1n
n1 2n
12k1
2
∞
1
∞
1
−
∞
1
−1
2k1
n1 n
− 32k1 n1 2n2k1
22k1 − 1
1
1
−
4k1 ζ 2k 1,
ζ2k 1 − 1.
4
2
22k1
n1 4n
3.14
By 3.13 and 3.14, we obtain the following.
Corollary 3.2. For n ∈ N,
E2n
1
22n 1 − 22n1 ζ2n 1 −1n
π 2n1 22n .
ζ 2n 1,
4
22n!
3.15
By simple calculation, we easily see that
i tan x eix − e−ix
2
4
1 − 2ix
.
eix e−ix
e − 1 e4ix − 1
3.16
Thus, we have
∞ −1n B 4n 1 − 4n
2xi
4xi
2n
x2n .
x tan x −xi 2xi
−
2n!
e − 1 e4xi − 1 n1
3.17
From 3.17, we can easily derive
∞ −1n1 4n1 1 − 4n1 B
2n2 2n1
x
tan x .
2n
2!
n0
3.18
By 3.3, we also see that
i tan x 1 −
∗
∞
E2n1
2
i
22n1 −1n1 x2n1 .
2n 1!
e2ix 1
n0
3.19
Thus, we have
tan x ∗
∞
E2n1
22n1 −1n1 x2n1 .
2n
1!
n0
3.20
By 3.18 and 3.20, we obtain the following.
Theorem 3.3. For n ∈ N,
∗
−1n−1 2π2n E2n−1
ζ2n ,
42n − 1! 1 − 4n
where En∗ are the first-kind Euler numbers.
3.21
Taekyun Kim
9
It is easy to see that
∞
1
k1 2k
12n
−1n 2π2n ∗
1
1 − n ζ2n n1
.
E
4
4 2n − 1! 2n−1
3.22
Therefore, we obtain the following corollary.
Corollary 3.4. For n ∈ N,
∞
1
k1 2k
12n
−1n 2π2n ∗
.
E
4n1 2n − 1! 2n−1
3.23
Now we try to give the new value of the Euler zeta function at positive integers. From
the definition of the Euler zeta function, we note that
ζE s 2
∞
−1n
n1
ns
−2
∞
1
1
s s−1 ζs,
2
2n
1
n0
s ∈ C.
3.24
By 3.24, Theorem 3.3, and Corollary 3.4, we obtain the following theorem.
Theorem 3.5. For n ∈ N,
−1n−1 π 2n 2 − 4n ∗
ζE 2n E2n−1 .
22n − 1! 1 − 4n
3.25
Remark 3.6. We note that ζ2 π 2 /6, ζE 2 −π 2 /6, ζ4 π 4 /90 and ζE 4 −7π 4 /360 . . ..
For q ∈ C with |q| < 1, s ∈ C, q-ζ-function is defined by
ζq s s
∞
qn
1 1 − q
s −
s − 1 log q
n1 nq
3.26
cf. 10, 14. Note that ζq s is analytic continuation in C with only one simple pole at s 1,
and
ζq 1 − k −
βk,q
,
k
where k is a positive integer
3.27
cf. 14. By simple calculation, we easily see that
∞ θ 2j1 n2j1
∞
−1n qn q
n1
n2k1
q
j0
2j 1!
k−1
k−1
1 − q2k−2j
1 − q2k−2j
1 1 −
log q j0 2k − 2j − 12j 1! log q j0 2k − 2j − 12j 1!
k−1
∞
q θ 2k1
−1j θ 2j1
θ 2j1 −1j H2j−2k,q −q−1
2
−ζq 2k−2jζq2 2k−2j 2k−2j −
−1k
,
2j 1!
1q 2k1!
2j 1!
1q
2
j0
jk1
q
3.28
10
Abstract and Applied Analysis
where Hn,q −q are Carlitz’s q-Euler numbers with limq→1 Hn,q −q En∗ cf. 6, 21, 22. If q→1,
then we have
∞
−1n
n1
n2k1
sinnθ 2k−2j
k−1
2
−1j 2j1
1 θ 2k1
k−j1 2π
θ
B
−1k .
−
1
×
−1
−
2k−2j
2k−2j
2j
1!
2
·
2k
−
2j!
2
2k
1!
2
j0
3.29
For k ∈ N, and θ π/2, it is easy to see that
∞
−1n
n1 2n
− 12k1
k−1 −1k π 2k1 22k−2j − 2 B
2k−2j
j0
2j 1!2k − 2j!22j2
−
π 2k1 −1k
.
2k 1!22k2
3.30
From 3.30 and Theorem 3.1, we can also derive the following equation:
k−1 −1k−1 π 2k1 22k−2j − 2 B
2k−2j
j0
2j 1!2k − 2j!22j2
2k1
π
π 2k1 −1k
k E2k
−1
.
22k! 2
2k 1!22k2
3.31
1
E2k
− 2k2
.
2k2
2k 1!2
2
2k!
3.32
Thus, we have
k−1
j0 2j
22k−2j − 2 B2k−2j
1!2k −
2j!22j2
Acknowledgments
The present research has been conducted by the research grant of Kwangwoon University in
2008. The author wishes to express his sincere gratitude to the referees and Professor Lance L.
Littlejohn for their valuable suggestions and comments.
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