Hindawi Publishing Corporation 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,
[email protected] 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
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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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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
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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
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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
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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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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
Discrete Dynamics in Nature and Society 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
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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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7 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. 8 T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008. 9 T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008. 10 T. Kim, “On explicit formulas of p-adic q-L-functions,” Kyushu Journal of Mathematics, vol. 48, no. 1, pp. 73–86, 1994. 11 T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009. 12 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. 13 T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 14–50, 2011. 14 A. Kudo, “A congruence of generalized Bernoulli number for the character of the first kind,” Advanced Studies in Contemporary Mathematics, vol. 2, pp. 1–8, 2000. 15 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009. 16 S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009. 17 C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010. 18 I. Buyukyazici, “On Generalized q-Bernstein Polynomials,” Global Journal of Pure and Applied Mathematics, vol. 6, no. 3, pp. 1331–1348, 2010. 19 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251– 278, 2008.
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