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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 856132, 11 pages
doi:10.1155/2011/856132
Research Article
Some New Identities on the Bernoulli and
Euler Numbers
Dae San Kim,1 Taekyun Kim,2 Sang-Hun Lee,3
D. V. Dolgy,4 and Seog-Hoon Rim5
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, Kwangwoon University, Seoul 139-701, Republic of Korea
4
Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea
5
Department of Mathematics Education, Kyungpook National University,
Taegu 702-701, Republic of Korea
2
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 6 October 2011; Revised 31 October 2011; Accepted 31 October 2011
Academic Editor: Lee-Chae Jang
Copyright q 2011 Dae San 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 give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic
integral on Zp and reflection symmetric properties of Bernoulli and Euler polynomials.
1. Introduction
Let p be a fixed prime number. Throughout this paper Zp , Qp , and Cp will denote the ring of
p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic
closure of Qp . Let UDZp be the space of uniformly differentiable functions on Zp . For f ∈
UDZp , the bosonic p-adic integral on Zp is defined by
I f pN −1
1 N
fxdμx lim
fxμ x p Zp lim N
fx.
N →∞
N →∞p
Zp
x0
x0
N
p
−1
1.1
From 1.1, we note that
I f1 I f f 0,
where f1 x fx 1,
1.2
2
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see 1. As is well known, the ordinary Bernoulli polynomials are defined by the generating
function as follows:
Ft, x ∞
tn
t
xt
Bxt
,
e
e
B
x
n
n!
et − 1
n0
1.3
see 1–19, where we use the technical notation by replacing Bn x by Bn xn ≥ 0,
symbolically. In the special case, x 0, Bn 0 Bn are called the n-th ordinary Bernoulli
numbers. That is, the generating function of ordinary Bernoulli numbers is given by
Ft Ft, 0 ∞
tn
t
Bn ,
t
e − 1 n0 n!
1.4
see 1–19. From 1.4, we can derive the following relation:
B 1n − Bn δ1,n ,
B0 1,
1.5
see 1, 10, where δ1,n is the Kronecker symbol.
By 1.3 and 1.4, we easily get
Bn x n
n
l0
l
Bl x
n−l
n
n
l
l0
Bn−l xl .
1.6
By 1.2 and 1.3, we easily get
Zp
exyt dμ y ∞
tn
t
xt
,
e
B
x
n
n!
et − 1
n0
1.7
see 1, 10. From 1.7, we can derive Witt’s formula for the n-th Bernoulli polynomials as
follows:
n x y dμ y Bn x,
Zp
where n ∈ Z ,
1.8
see 11. By 1.1 and 1.8, we easily see that
n y 1 − x dμ y −1n
Zp
Zp
n y x dμ y .
1.9
Thus, by 1.8 and 1.9, we get reflection symmetric relation for the Bernoulli polynomials
as follows:
Bn 1 − x −1n Bn x
where n ∈ Z .
1.10
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3
The ordinary Euler polynomials are defined by the generating function as follows:
Fe t, x ∞
tn
2
xt
.
e
E
x
n
n!
et 1
n0
1.11
with the usual convention about replacing En x by En x see 8, 9. In the special case,
x 0, En 0 En are called the n-th Euler numbers see 8, 9.
From 1.11, we note that
∞
2
2
tn
n
xt
−1−xt
,
e
e
E
−
x
−1
1
n
n!
et 1
1 e−t
n0
1.12
By comparing the coefficients on both sides of 1.11 and 1.12, we obtain the following
reflection symmetric relation for Euler polynomials as follows:
En x −1n En 1 − x,
where n ∈ Z .
1.13
The equations 1.10 and 1.13 are useful in deriving our main results in this paper.
For n, k ∈ Z , the Bernstein polynomials are defined by
Bk,n x n
xk 1 − xn−k ,
k
1.14
see 13. By 1.14, we easily get Bk,n x Bn−k,n 1 − x.
In this paper we consider the p-adic integrals for the Bernoulli and Euler polynomials.
From those p-adic integrals, we derive some new identities on the Bernoulli and Euler
numbers.
2. Identities on the Bernoulli and Euler Numbers
First, we consider the p-adic integral on Zp for the nth ordinary Bernoulli polynomials as
follows:
I1 Zp
Bn xdμx l0
n
n
l0
l
n
n
l
Bn−l
Zp
xl dμx
2.1
where n ∈ Z .
Bn−l Bl ,
On the other hand, by 1.3 and 1.10, one gets
I1 −1n
Zp
Bn 1 − xdμx.
2.2
4
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From 1.5, 1.6, 1.8, and 2.2, one notes that
I1 −1
−1
n
n
n
n
l0
l
l0
l
Bn−l
1 − xl dμx
Zp
n
n
2.3
Bn−l l Bl δ1,l n
−1 nBn−l 1 −1
n
n
n
l0
l
Bn−l Bl −1n nBn−l .
Equating 2.1 and 2.3, one gets
1 −1
n1
n
n
Bn−l Bl −1n nδ1,n−l Bn−1 −1n nBn−1
l
l0
n
2.4
n
2−1 nBn−l −1 nδ1,n−1 .
Let n ∈ N with n ≡ 1 mod 2. Then, by 2.4, one has
2n−1
2n − 1
B2n−1−l Bl −2n − 1B2n−2 .
l
l0
2.5
Therefore, by 2.4 and 2.5, we obtain the following theorem.
Theorem 2.1. For n ∈ N, one has
n1
1 −1
n
n
l
l0
Bn−l Bl 2−1n nBn−1 −1n nδ1,n−1 .
2.6
In particular,
2n−1
2n − 1
l
l0
B2n−1−l Bl −2n − 1B2n−2 .
2.7
By the same motivation, let us also consider the p-adic integral on Zp for Euler polynomials
as follows:
I2 Zp
En xdμx l0
n
n
l0
l
n
n
En−l Bl ,
l
En−l
where n ∈ Z .
Zp
xl dμx
2.8
Discrete Dynamics in Nature and Society
5
On the other hand, by 1.12 and 1.13, one gets
I2 −1
−1
n
Zp
En 1 − xdμx −1
n
n
En−l
l
l0
n
n
n
Zp
1 − xl dμx
2.9
En−l l Bl δ1,l l
l0
n
n
n−1 En−l 1 −1
n
n
n
En−l Bl −1n nEn−l .
l
l0
From 1.12 and the definition of Euler numbers, one has
En x n
n
El x
l
l0
n−l
n
n
l0
l
En−l xl E xn ,
E 1n En 2δ0,n ,
E0 1,
2.10
2.11
see 8, 9 with the usual convention of replacing En by En . By 2.9, 2.10, and 2.11, one
gets
n
n
I2 n−1 2δ0,n−1 − En−1 −1 nEn−1 −1
n
n
n
l0
l
En−l Bl .
2.12
Equating 2.8 and 2.12, one has
1 −1
n−1
n
n
En−l Bl 2n−1n δ0,n−1 .
l0
l
2.13
Therefore, by 2.13, we obtain the following theorem.
Theorem 2.2. For n ∈ N ∪ {0}, one has
1 −1
n−1
n
n
l
l0
En−l Bl 2−1n nδ0,n−1 .
2.14
In particular,
2n1
l0
2n 1
l
E2n1−l Bl 0,
for n ∈ N.
2.15
6
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Let us consider the following p-adic integral on Zp for the product of Bernoulli and
Euler polynomials as follows:
Bm xEn xdμx
I3 Zp
n
m m
n
k0 0
k
k0 0
k
Bm−k En−
Zp
xk xdμx
2.16
n
m m
n
Bm−k En− Bk .
On the other hand, by 1.10 and 1.13, one gets
I3 −1mn
Bm 1 − xEn 1 − xdμx
Zp
−1
mn
m n
m
n
k0 0
−1
mn
k
m n
m
n
k0 0
k
Bm−k En−
Zp
1 − xk dμx
Bm−k En− k Bk δ1,k 2.17
−1mn mBm−1 1En 1 −1mn nBm 1En−1 1
m n
m
n
mn
Bm−k En− Bk −1mn mBm−1 En nBm En−1 .
−1
k
k0 0
Equating 2.16 and 2.17, one gets
n
m m
n
mn1
Bm−k En− Bk
1
−1
k
k0 0
−1mn mBm−1 δ1,m−1 2δ0,n − En 2.18
−1mn nBm δ1,m 2δ0,n−1 − En−1 −1mn nBm En−1 mBm−1 En .
For n ∈ N, by 2.18, one gets
−1
m1
2n
m m
2n
Bm−k E2n− Bk
1
k
k0 0
−1m1 2nBm δ1,m E2n−1 −1m 2nBm E2n−1 −1m1 2nδ1,m E2n−1 .
Therefore, by 2.19, one obtains the following theorem.
2.19
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7
Theorem 2.3. For n ∈ N, one has
−1
m1
2n
m m
2n
Bm−k E2n− Bk −1m1 2nδ1,m E2n−1 .
1
k
k0 0
2.20
In particular, for m ∈ N, one has
2n
2m1
2m 1
k0 0
k
2n
B2m1−k E2n− Bk 0.
2.21
By the same motivation, we consider the p-adic integral on Zp for the product of
Bernoulli and Bernstein polynomials as follows:
I4 Zp
where m, n, k ∈ N ∪ {0}.
Bm xBk,n xdμx
2.22
From 1.6 and 1.14, one gets
I4 m
m
x Bk,n xdμx
Bm−
Zp
0
m
n m
Bm−
xk 1 − xn−k dμx
k 0 Zp
n−k
m n m
n−k
j
Bm− Bkj .
−1
k 0 j0
j
2.23
On the other hand,
I4 −1m
−1
−1
m
Zp
Bm 1 − xBn−k,n 1 − xdμx
k
m n k
m
0 j0
n
−1
j
m
k
j
Bm− n − k j Bn−kj δ1,n−kj
m
n
mBm−1 1δ0,k − −1
n − kBm 1δ0,k −1
k
k
k
m n m
k
m
j
Bm− Bn−kj
−1
−1
k 0 j0
j
−1
m
n
k
mBm−1 − kBm δn,k −1
m
m
n
k
mBm 1kδ0,k−1
n
k
Bm δn,k1 .
2.24
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Equating 2.23 and 2.24, one gets
m n−k
−1
−1j
m
m
n−k
0 j0
Bm− Bkj
j
n − kBm 1 mBm−1 1δ0,k − kBm 1δ0,k−1 mBm−1 − kBm δn,k
k
m m
k
j
Bm− Bn−kj .
Bm δn,k1 −1
j
0 j0
2.25
By 2.25, we obtain the following theorem.
Theorem 2.4. For n, m ∈ N, one has
2n
2m −1j
2m
2n
0 j0
j
B2m− Bj
2m
2m
B2m− B2n .
2nB2m 0
2.26
Now, we consider the p-adic integral on Zp for the product of Euler and Bernstein
polynomials as follows:
I5 Zp
Em xBk,n xdμx
m
m
Em−
0
n−k
m n k
Zp
−1
j
x Bk,n xdμx
m
n−k
0 j0
j
Em− Bkj .
On the other hand, by 1.13 and 1.14, one gets
I5 −1m
Zp
Bn−k,n 1 − xEm 1 − xdμx
k
m n m
k
j
Em−
−1
−1
1 − xn−kj dμx
k 0 j0
j
Zp
m
k
m n m
k j
−1
n − k j Bn−kj δ1,n−kj Em−
−1
k 0 j0
j
m
2.27
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m
−1 n − k
n
k
Em 1δ0,k −1
−1
m
k
m n −1
m
n
9
m
n
mEm−1 1δ0,k − −1
k
m
n
k
Em 1kδ0,k−1
m
k
Em− Bn−kj
−1
k 0 j0
j
j
δn,k1 Em δn,k mEm−1 − kEm .
k
Equating 2.27 and 2.28, one gets
m n−k
−1j
−1
m
0 j0
2.28
m
n−k
Em− Bkj
j
n − kEm 1δ0,k mδ0,k Em−1 1 − kEm 1δ0,k−1
k
m m
k
j
Em− Bn−kj
−1
j
0 j0
2.29
δn,k1 Em mEm−1 − kEm δn,k .
Therefore, by 2.11 and 2.29, we obtain the following theorem.
Theorem 2.5. For n, m ∈ N, one has
2n
2m 2m
2n
j
E2m− Bj −2mE2m−1 B2m2n .
−1
j
0 j0
2.30
Finally, we consider the p-adic integral on Zp for the product of Euler, Bernoulli, and Bernstein
polynomials as follows:
Br xEs xBk,n xdμx
I6 Zp
s
r r
s
n Br− Es−j
xkj 1 − xn−k dμx
j
k 0 j0 Zp
s r n−k
n k
0 j0 i0
−1
i
2.31
r
s
n−k
j
i
Br− Es−j Bkij .
On the other hand, by 1.10, 1.13, and 1.14, one gets
I6 −1rs
Br 1 − xEs 1 − xBn−k,n 1 − xdμx
Zp
s r k
n r
s
k
rs
i
Br− Es−j
−1
−1
1 − xn−kij dμx.
k 0 j0 i0
j
i
Zp
2.32
10
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Equating 2.31 and 2.32, we easily see that
−1
rs
r
s
n−k
Br− Es−j Bkij
−1
j
i
0 j0 i0
r s n−k
i
s r k
r
s
k i
n − k i j Bn−kij δ1,n−kij Br− Es−j
−1
j
i
0 j0 i0
n − kBr 1Es 1δ0,k rBr−1 1δ0,k Es 1 sBr 1Es−1 1δ0,k
s r k
r
s
k
i
Br− Es−j Bn−kij
− kBr 1Es 1δ0,k−1 −1
j
i
0 j0 i0
δn,k1 Br Es rBr−1 Es sBr Es−1 − kBr Es δn,k .
2.33
Therefore, by 1.5 and 2.11, we obtain the following theorem.
Theorem 2.6. For r, n, s ∈ N, one has
2s 2r 2n
2r
2s
2n
i
B2r− E2s−j Bij
−1
j
i
0 j0 i0
−2sB2r E2s−1 r
2r
0
2l
B2r−2l B2n2l2s − r
s
j1
2s
2j − 1
2.34
E2s−2j1 B2n2r2j−2 .
Acknowledgments
The authors express their sincere gratitude to the referees for their valuable suggestions and
comments. This paper is supported in part by the Research Grant of Kwangwoon University
in 2011.
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