Some New Identities on the Bernoulli and Euler Numbers

1 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea 2 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


Introduction
Let p be a fixed prime number.Throughout this paper Z p , Q p , and C p will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p .Let UD Z p be the space of uniformly differentiable functions on Z p .For f ∈ UD Z p , the bosonic p-adic integral on Z p is defined by From 1.1 , we note that 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: 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 . By 1.14 , we easily get B k,n x B n−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.

Identities on the Bernoulli and Euler Numbers
First, we consider the p-adic integral on Z p for the nth ordinary Bernoulli polynomials as follows: where n ∈ Z .

2.1
On the other hand, by 1.3 and 1.10 , one gets From 1.5 , 1.6 , 1.8 , and 2.2 , one notes that

2.6
In particular, By the same motivation, let us also consider the p-adic integral on Z p for Euler polynomials as follows: where n ∈ Z .

2.8
On the other hand, by 1.12 and 1.13 , one gets 2.9 From 1.12 and the definition of Euler numbers, one has Equating 2.8 and 2.12 , one has Therefore, by 2.13 , we obtain the following theorem.

2.14
In particular,

2.15
Let us consider the following p-adic integral on Z p for the product of Bernoulli and Euler polynomials as follows:

2.16
On the other hand, by 1.10 and 1.13 , one gets 2.17 Equating 2.16 and 2.17 , one gets

2.18
For n ∈ N, by 2.18 , one gets

2.19
Therefore, by 2.19 , one obtains the following theorem.

2.20
In particular, for m ∈ N, one has By the same motivation, we consider the p-adic integral on Z p for the product of Bernoulli and Bernstein polynomials as follows:

2.22
From 1.6 and 1.14 , one gets

2.23
On the other hand,

2.25
By 2.25 , we obtain the following theorem.

2.26
Now, we consider the p-adic integral on Z p for the product of Euler and Bernstein polynomials as follows:

2.27
On the other hand, by 1.13 and 1.14 , one gets

2.29
Therefore, by 2.11 and 2.29 , we obtain the following theorem.

2.30
Finally, we consider the p-adic integral on Z p for the product of Euler, Bernoulli, and Bernstein polynomials as follows:

2.31
On the other hand, by 1.10 , 1.13 , and 1.14 , one gets

2.32
Equating 2.31 and 2.32 , we easily see that

Theorem 2 . 6 . 1 E
n−k i j δ 1,n−k i j B r− E s−j n − k B r 1 E s 1 δ 0,k rB r−1 1 δ 0,k E s 1 sB r 1 E s−1 1 δ 0,k − kB r 1 E s 1 δ 0,kE s−j B n−k i j δ n,k 1 B r E s rB r−1 E s sB r E s−1 − kB r E s δ n,k .2.33Therefore, by 1.5 and 2.11 , we obtain the following theorem.For r, n, s ∈ N, one has 2s−2j 1 B 2n 2r 2j−2 .