Arithmetic Identities Involving Bernoulli and Euler Numbers

Let p be a fixed odd 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, respectively. The p-adic norm is normalized so that |p|p 1/p. Let N be the set of natural numbers and Z N ∪ {0}. Let UD Zp be the space of uniformly differentiable functions on Zp. For f ∈ UD Zp , the bosonic p-adic integral on Zp is defined by


Introduction
Let p be a fixed odd 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 , respectively.The p-adic norm is normalized so that |p| p 1/p.Let N be the set of natural numbers and Z N ∪ {0}.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 and the fermionic p-adic integral on Z p is defined by Kim as follows see 1-8 :

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The Euler polynomials, E n x , are defined by the generating function as follows see 1-16 : In the special case, x 0, E n 0 E n is called the nth Euler number.By 1.3 and the definition of Euler numbers, we easily see that with the usual convention about replacing E l by E l see 10 .Thus, by 1.3 and 1.4 , we have where δ k,n is the Kronecker symbol see 9, 10, 17-19 .From 1.2 , we can also derive the following integral equation for the fermionic p-adic integral on Z p as follows: where f 1 x f x 1 and f 0 df x /dx | x 0 .By 1.12 , we have Thus, by 1.13 , we can derive the following Witt's formula for the Bernoulli polynomials: In 19 , it is known that for k, m ∈ Z , max{k,m} where The purpose of this paper is to give some arithmetic identities involving Bernoulli and Euler numbers.To derive our identities, we use the properties of p-adic integral equations on Z p .

Arithmetic Identities for Bernoulli and Euler Numbers
Let us take the bosonic p-adic integral on Z p in 1.15 as follows:

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On the other hand, we get

2.2
By 2.1 and 2.2 , we get max{k,m}

2.3
Therefore, by 2.3 , we obtain the following theorem.

2.4
Now we consider the fermionic p-adic integral on Z p in 1.15 as follows:

2.5
On the other hand, we get

2.6
By 2.5 and 2.6 , we get max{k,m}

2.7
Therefore, by 2.7 , we obtain the following theorem.

2.8
Replacing x by 1 − x in 1.15 , we have the identity: max{k,m}

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Let us take the bosonic p-adic integral on Z p in 2.9 as follows:

2.10
International Journal of Mathematics and Mathematical Sciences 7 On the other hand, we see that

2.11
By 2.10 and 2.11 , we get max{k,m}

2.12
Therefore, by 2.12 , we obtain the following theorem.

2.13
We consider the fermionic p-adic integral on Z p in 2.9 as follows: International Journal of Mathematics and Mathematical Sciences max{k,m}

2.14
On the other hand, we get

2.15
By 2.14 and 2.15 , we obtain the following theorem. 2.16

Theorem 2 . 4 .
For k, m ∈ Z , one has -19, with the usual convention about replacing B l by B l .By 1.9 and 1.10 , we easily see that From 1.1 , we can derive the following integral equation on Z p :