Identities on the Bernoulli and Genocchi Numbers and Polynomials

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 the algebraic closure of Qp. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic norm on Cp is normalized so that |p|p p−1. Let C Zp be the space of continuous functions on Zp. For f ∈ C Zp , the fermionic p-adic integral on Zp is defined by Kim as follows:


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 the algebraic closure of Q p .Let N be the set of natural numbers and Z N ∪ {0}.The p-adic norm on C p is normalized so that |p| p p −1 .Let C Z p be the space of continuous functions on Z p .For f ∈ C Z p , the fermionic p-adic integral on Z p is defined by Kim as follows: From 1.2 , we have where the symbol δ 0,n is the Kronecker symbol see 4, 5 .Thus, by 1.5 and 1.7 , we get x y n dμ −1 y .1.9 By 1.6 and 1.9 , we see that Thus, by 1.10 , we get From 1.5 and 1.8 , we have 13 see 6 .Thus, by 1.12 and 1.13 , we get reflection symmetric formula for the Bernoulli polynomials as follows: 15 see 6, 9, 12 .From 1.14 and 1.15 , we can also derive the following identity: In this paper, we investigate some properties of the fermionic p-adic integrals on Z p .By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics.

Identities on the Bernoulli and Genocchi Numbers and Polynomials
Let us consider the following fermionic p-adic integral on Z p as follows:

International Journal of Mathematics and Mathematical Sciences
On the other hand, by 1.14 and 1.15 , we get

2.2
Equating 2.1 and 2.2 , we obtain the following theorem.
Theorem 2.1.For n ∈ Z , one has By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers.Now, we consider the fermionic p-adic integral on Z p for the polynomials as follows:

2.4
On the other hand, by 1.8 , 1.10 , and 1.11 , we get

2.5
Equating 2.4 and 2.5 , we obtain the following theorem.
Theorem 2.2.For n ∈ Z , one has Let us consider the fermionic p-adic integral on Z p for the product of B n x and G n x as follows:

2.7
On the other hand, by 1.10 and 1.14 , we get

2.8
International Journal of Mathematics and Mathematical Sciences By 2.7 and 2.8 , we easily see that 2.9 Therefore, by 2.9 , we obtain the following theorem.

2.11
Let us consider the fermionic p-adic integral on Z p for the product of the Bernoulli polynomials and the Bernstein polynomials.For n, k ∈ Z , with 0

2.12
International Journal of Mathematics and Mathematical Sciences 7 On the other hand, by 1.14 and 2.12 , we get

2.13
Equating 2.12 and 2.13 , we see that

2.14
Thus, from 2.14 , we obtain the following theorem.
Theorem 2.5.For n, m ∈ N, one has Finally, we consider the fermionic p-adic integral on Z p for the product of the Euler polynomials and the Bernstein polynomials as follows: International Journal of Mathematics and Mathematical Sciences

2.16
On the other hand, by 1.10 and 2.12 , we get

2.18
Therefore, by 2.18 , we obtain the following theorem.