Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 486158, 12 pages
doi:10.1155/2012/486158
Research Article
Some Identities on Bernoulli and Euler Numbers
D. S. Kim,1 T. Kim,2 J. Choi,3 and Y. H. Kim3
1
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 15 November 2011; Accepted 23 December 2011
Academic Editor: Delfim F. M. Torres
Copyright q 2012 D. S. 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.
Recently, Kim introduced the fermionic p-adic integral on Zp . By using the equations of the
fermionic and bosonic p-adic integral on Zp , we give some interesting identities on Bernoulli and
Euler numbers.
1. Introduction/Preliminaries
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will denote the
ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic
closure of Qp , respectively. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic
absolute value | · |p is normally defined by |p|p 1/p.
Let UDZp be the space of uniformly differentiable functions on Zp and CZp the
space of continuous function on Zp . For f ∈ CZp , the fermionic p-adic integral on Zp is
defined by Kim as follows:
I−1 f Zp
fxdμ−1 x lim
N
−1
p
N →∞
fx−1x ,
see 1.
1.1
x0
The following fermionic p-adic integral equation on Zp is well known see 1–3:
I−1 f1 I−1 f 2f0,
where f1 x fx 1.
1.2
2
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From 1.1 and 1.2, we can derive the generating function of Euler polynomials as
follows:
Zp
exyt dμ−1 y et
∞
tn
2
ext En x ,
n!
1
n0
1.3
where En x is the nth ordinary Euler polynomial see 1–4. In the special case, x 0,
En 0 En is called the nth ordinary Euler number.
By 1.3, we get Witt’s formula for the nth Euler polynomial as follows:
Zp
n
x y dμ−1 y En x,
for n ∈ Z .
1.4
Thus, by 1.4, we have
n
En x E x n n
l0
l
xn−l El ,
1.5
with the usual convention about replacing En by En see 5, 6. From 1.3, we note that
E 1n En 2δ0,n ,
1.6
where δk,n is the Kronecker symbol see 3. By 1.2 and 1.4, we get
Zp
n
x y 1 dμ−1 y Zp
n
x y dμ−1 y 2xn .
1.7
Thus, by 1.4 and 1.7, we have
for n ∈ Z .
En x 1 En x 2xn ,
1.8
Equation 1.8 is equivalent to
xn En x n−1 1
n
El x.
2 l0 l
1.9
From 1.6, we can derive the following equation:
En 2 2 − En 1 2 En − 2δ0,n ,
for n ∈ Z .
1.10
For f ∈ UDZp , the bosonic p-adic integral on Zp is defined by
I1 f p −1
1 fxdμ1 x lim N
fx,
N →∞p
Zp
x0
N
see 4.
1.11
Discrete Dynamics in Nature and Society
3
From 1.11, we can easily derive the following I1 -integral equation:
I1 f1 I f f 0,
see 4, 7, 8,
1.12
where f1 x fx 1 and f 0 dfx/dx|x0 .
It is well known that the Bernoulli polynomial can be represented by the bosonic p-adic
integral on Zp as follows:
Zp
exyt dμ1 y ∞
tn
t
xt
e
Bn x ,
t
n!
e −1
n0
1.13
where Bn x is called the nth Bernoulli polynomial see 4, 7–13. In the special case, x 0,
Bn 0 Bn is called the nth Bernoulli number. By the definition of Bernoulli numbers and
polynomials, we get
Bn x n n
n n−l
x y dμ1 y x Bl .
l
l0
Zp
1.14
Thus, by 1.13 and 1.14, we see that
B0 1,
B 1n − Bn δ1,n ,
1.15
with the usual convention about replacing Bn by Bn see 1–22.
By 1.11, we easily get
Zp
n
1 − x y dμ1 y −1n
Zp
n
x y dμ1 y .
1.16
From 1.13, 1.14, and 1.16, we have
Bn 1 − x −1n Bn x
for n ∈ Z .
1.17
By 1.15, we get
Bn 2 n Bn 1 n Bn δ1,n .
1.18
Thus, by 1.17 and 1.18, we have
−1n Bn −1 Bn 2 n Bn δ1,n ,
see 4.
1.19
dμ1 y n 1xn .
1.20
From 1.12 and 1.13, we get
Zp
x1y
n1
dμ1 y −
Zp
xy
n1
4
Discrete Dynamics in Nature and Society
Thus, by 1.13 and 1.20, we have
for n ∈ Z .
Bn1 x 1 − Bn1 x n 1xn
1.21
Equation 1.21 is equivalent to the following equation:
xn n 1 n1
Bl x
l
n 1 l0
for n ∈ Z .
1.22
In this paper we derive some interesting and new identities for the Bernoulli and Euler
numbers from the p-adic integral equations on Zp .
2. Some Identities on Bernoulli and Euler Numbers
From 1.1, we note that
Zp
n
1 − x y dμ−1 y −1n
Zp
n
x y dμ−1 y .
2.1
By 1.14 and 2.1, we get
En 1 − x −1n En x,
where n ∈ Z .
2.2
In the special case, x −1, we have
En 2 −1n En −1 2 En − 2δ0,n .
2.3
Let us consider the following fermionic p-adic integral on Zp as follows:
n 1 n1
x dμ−1 x Bl xdμ−1 x
l
n 1 l0
Zp
Zp
n
l n 1 n1 l
xk dμ−1 x
Bl−k
l
k
n 1 l0
Z
p
k0
2.4
l n 1 n1 l
Bl−k Ek .
l
k
n 1 l0
k0
Therefore, by 1.4 and 2.4, we obtain the following theorem.
Theorem 2.1. For n ∈ Z , one has
En l n 1 n1 l
Bl−k Ek .
l
k
n 1 l0
k0
2.5
Discrete Dynamics in Nature and Society
5
It is known that Bn x −1n Bn 1 − x. If we take the fermionic p-adic integral on
both sides of 1.22, then we have
n 1 n1
x dμ−1 x Bl xdμ−1 x
l
n 1 l0
Zp
Zp
n
n 1 n1
l
Bl 1 − xdμ−1 x
−1
l
n 1 l0
Zp
l n 1 n1
l
Bl−k
1 − xk dμ−1 x
−1l
l
k
n 1 l0
Zp
k0
2.6
l n 1 n1
l
l
Bl−k −1k Ek −1.
−1
l
k
n 1 l0
k0
From 2.2 and 2.6, we note that
Zp
xn dμ−1 x l n 1 n1
l
Bl−k Ek 2
−1l
l
k
n 1 l0
k0
l n 1 n1
l
B 2 Ek − 2δ0,k −1l
l
k l−k
n 1 l0
k0
n l 1 n1
l
l
B E − 2Bl
−1 2Bl 1 l
k l−k k
n 1 l0
k0
2.7
n l
1 n1
l
l
B E 2δ1,l .
−1
l
k l−k k
n 1 l0
k0
Therefore, by 1.4 and 2.7, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , one has
n l
1 n1
l
l
En B E 2δ1,l .
−1
l
k l−k k
n 1 l0
k0
2.8
Corollary 2.3. For n ∈ N, one has
n l
1 n1
l
l
2 En B E .
−1
l
k l−k k
n 1 l0
k0
2.9
6
Discrete Dynamics in Nature and Society
Let us take the bosonic p-adic integral on both sides of 1.9 as follows:
n−1 1
n
En x El x dμ1 x
2 l0 l
x dμ1 x n
Zp
Zp
n n
l
l0
n n
l
l0
En−l
n−1 1
n l l
x dμ1 x xk dμ1 x
El−k
2 l0 l k0 k
Zp
Zp
En−l Bl l
2.10
n−1 1
n l l
El−k Bk .
2 l0 l k0 k
Thus, by 1.14 and 2.10, we obtain the following theorem.
Theorem 2.4. For n ∈ Z , one has
Bn n n−1 1
n
n l l
En−l Bl El−k Bk .
l
2 l0 l k0 k
l0
2.11
On the other hand, by 2.2 and 2.10, we get
x dμ1 x −1
n
Zp
n
n−1 1
n
l
En 1 − xdμ1 x El 1 − xdμ1 x
−1
2 l0 l
Zp
Zp
n n
−1
En−l
1 − xl dμ1 x
l
Zp
l0
n
l n−1 1
n
l
El−k
−1l
1 − xk dμ1 x
k
2 l0 l
Zp
k0
−1n
n l n−1 1
n
n
l
En−l −1l Bl −1 El−k −1k Bk −1
−1l
l
l
k
2
l0
l0
k0
−1n
n−1 n l 1
n
n
l
En−l Bl 2 El−k Bk 2
−1l
l
l
k
2
l0
l0
k0
−1n
n l n−1 1
n
n
l
En−l l Bl δ1,l El−k k Bk δ1,k −1l
l
l
k
2 l0
l0
k0
Discrete Dynamics in Nature and Society
n
−1 nEn−1 1 −1
n
n n
l0
7
n−1 1
n
En−l Bl −1 nEn−1 −1l lEl−1 1
l
2 l0 l
n
l n−1 n−1 1
1
n
l
n
El−k Bk −1l
−1l lEl−1
k
l
2 l0 l
2
k0
l1
−1n n2 En−1 − 2δ0,n−1 −1n
n n
En−l Bl −1n nEn−1
l
l0
l n−1 n−1 1
1
n
n
l
El−k Bk
−1l l2 El−1 − δ0,l−1 −1l
l
l
k
2 l0
2 l0
k0
n−1 1
n
−1l lEl−1 ,
2 l1 l
2.12
where n ∈ N with n ≥ 2. Therefore, by 2.12, we obtain the following theorem.
Theorem 2.5. For n ∈ N with n ≥ 2, one has
B2n−1 −
2n−1
2n − 1
2n − 1
− 2n − 1E2n−2 −1 −
E2n−1−l Bl
l
2
l0
l 2n − 1
1 2n−2
l
l
El−k Bk .
−1
l
k
2 l0
k0
2.13
By 1.9 and 1.22, we get
Zp
xm yn dμ−1 xdμ1 y
Zp
m n−1 1
1 m1
n
En y Bk x
El y dμ−1 xdμ1 y
k
l
m 1 k0
2 l0
m 1 m1
Bk xEn y dμ−1 xdμ1 y
k
m 1 k0
Zp
m
n−1
m 1n
1
Bk xEl y dμ−1 xdμ1 y
k
l
2m 1 k0 l0
Zp
m k n
1
m1
k
n
B E B E
k
l
p k−l n−p p l
m 1 k0 l0 p0
n−1 k m l 1
m1
n
k
l
B E EB .
k
l
s
p k−s l−p s p
2m 1 k0 l0 s0 p0
Therefore, by 1.4, 1.14, and 2.14, we obtain the following theorem.
2.14
8
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Theorem 2.6. For m ∈ Z and n ∈ N, one has
Em Bn k n m 1 m1
k
n
B E B E
k
l
p k−l n−p p l
m 1 k0 l0 p0
n−1 k l m 1
m1
n
k
l
B E EB .
k
l
s
p k−s l−p s p
2m 1 k0 l0 s0 p0
2.15
It is easy to show that
m n−1 1 1
m1
n
En x dμ−1 x Bk x
El x dμ−1 x
k
m 1 k0
2 l0 l
Zp
k m n 1 m1
k
n
xij dμ−1 x
Bk−i En−j
k
i
j
m 1 k0 i0 j0
Zp
n−1 k m l 1
m1
n
k
l
xij dμ−1 x
Bk−i El−j
k
l
i
j
2m 1 k0 l0 i0 j0
Zp
Zp
x
mn
k m n 1 m1
k
n
Bk−i En−j Eij
k
i
j
m 1 k0 i0 j0
n−1 k m l 1
m1
n
k
l
Bk−i El−j Eij .
k
l
i
j
2m 1 k0 l0 i0 j0
2.16
Therefore, by 2.16, we obtain the following corollay.
Corollary 2.7. For m ∈ Z and n ∈ N, one has
Emn n k m 1 m1
k
n
Bk−i En−j Eij
k
i
j
m 1 k0 i0 j0
n−1 k m l 1
m1
n
k
l
B E E .
k
l
i
j k−i l−j ij
2m 1 k0 l0 i0 j0
2.17
For f ∈ CZp , p-adic analogue of Bernstein operator of order n for f is given by
n n k
k
n k
Bk,n x,
f
f
Bn f | x x 1 − xn−k k
n
n
k0
k0
2.18
where Bk,n x nk xk 1 − xn−k for n, k ∈ Z is called the Bernstein polynomial of degree n
see 8. From the definition of Bk,n x, we note that Bn−k,n 1 − x Bk,n x.
Discrete Dynamics in Nature and Society
9
Let us take the fermionic p-adic integral on Zp for the product of xm and Bk,n x as
follows:
m 1 m1
x Bk,n xdμ−1 x Bl xBk,n xdμ−1 x
l
m 1 l0
Zp
Zp
m
m l nk m1
l
xjk 1 − xn−k dμ−1 x
Bl−j
l
j
m 1 l0 j0
Zp
l m n−k
nk m1
l
n−k
xijk dμ−1 x
−1i Bl−j
l
j
i
m 1 l0 j0 i0
Zp
l m n−k
nk m1
l
n−k
Bl−j Eijk .
−1i
l
j
i
m 1 l0 j0 i0
2.19
From 2.18, we note that
Zp
xm Bk,n xdμ−1 x n
xmk 1 − xn−k dμ−1 x
k
Zp
n n−k n − k
j
xmkj dμ−1 x
−1
j
k j0
Zp
2.20
n n−k n − k
−1j Emkj .
j
k j0
Therefore, by 2.19 and 2.20, we obtain the following theorem.
Theorem 2.8. For m, n, k ∈ Z , one has
n−k n−k
j0
j
−1j Emkj l m n−k
1 m1
l
n−k
Bl−j Eijk .
−1i
l
j
i
m 1 l0 j0 i0
2.21
In particular,
m 1Emn
l m m1
l
Bl−j Ejn .
l
j
l0 j0
2.22
10
Discrete Dynamics in Nature and Society
By 1.17 and the symmetric property of Bk,n x, we get
Zp
xm Bk,n xdμ−1 x Zp
xm Bn−k,n 1 − xdμ−1 x
m
1 m1
Bl 1 − xBn−k,n 1 − xdμ−1 x
−1l
l
m 1 l0
Zp
l m k
nk l
k
il m 1
Bl−j
−1
1 − xijn−k dμ−1 x.
l
j
i
m 1 l0 j0 i0
Zp
2.23
From 1.4 and 2.2, we note that
Zp
1 − xn dμ−1 x −1n En −1 En 2 2 En − 2δ0,n .
2.24
By 2.23 and 2.24, we see that
Zp
xm Bk,n xdμ−1 x k
l m nk m1
l
k
Bl−j 2 Eijn−k − 2δ0,ijn−k .
−1il
l
j
i
m 1 l0 j0 i0
2.25
From 2.20 and 2.25, we have
n−k n−k
j
j0
l
m m
2δ0,k 2 m1
l
m1
Bl−j −
Bl δk,n
−1l
−1l
l
j
l
m 1 l0 j0
m 1 l0
−1j Emkj
l m k
1 m1
l
k
Bl−j Eijn−k
−1il
l
j
i
m 1 l0 j0 i0
l
m 2δ0,k 2 m1
l
Bm1 2 −1m Bm1 δk,n
Bl−j −
−1l
l
j
m 1 l0 j0
m1
l m k
1 m1
l
k
Bl−j Eijn−k .
−1il
l
j
i
m 1 l0 j0 i0
Therefore, by 1.19 and 2.26, we obtain the following theorem.
2.26
Discrete Dynamics in Nature and Society
11
Theorem 2.9. For m, n, k ∈ N with n ≥ k, one has
k
n−k m l 1 n−k
l
k
j
il m 1
Bl−j Eijn−k
−1
−1 Emkj j
l
j
i
m 1 l0 j0 i0
j0
2 −
Bm1 m 1 −1m Bm1 .
m1
2.27
In particular,
2m 2E2mn1 2 n
l 2m1
−1il
l0 j0 i0
2m 2
l
n
Bl−j Eij .
l
j
i
2.28
Acknowledgment
The first author was supported by National Research Foundation of Korea Grant funded by
the Korean Government 2011-0002486.
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