Integral Formulae of Bernoulli and Genocchi Polynomials

Recently, some interesting and new identities are introduced in the work of Kim et al. (2012). From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.


Introduction
As it is well known, the Bernoulli polynomials are defined by generating functions as follows: From 1.5 , we can derive the following equation: By the definition of Bernoulli and Genocchi numbers, we get the following recurrence formulae: where δ n,k is the Kronecker symbol see 2 .From 1.4 , 1.6 , and 1.7 , we note that From the identities of Bernoulli and Genocchi polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli and Genocchi polynomials.

Integral Formula of Bernoulli and Genocchi Polynomials
From 1.1 and 1.2 , we note that

2.1
By comparing the coefficients on the both sides of 2.1 , we obtain the following theorem.
Theorem 2.1.For n ∈ Z , one has From 1.1 and 1.2 , also notes that

2.3
By comparing the coefficients on the both sides of 2.3 , we obtain the following theorem.
Theorem 2.2.For n ∈ N, one has Let one take the definite integral from 0 to 1 on both sides of Theorem 2.1.For n ≥ 2, Therefore, by 2.3 , we obtain the following theorem.
Theorem 2.3.For n ∈ N with n ≥ 2, one has

p-Adic Integral on Z p Associated with Bernoulli and Genocchi Numbers
Let p be a fixed odd prime number.Throughout this section, Z p , Q p , and C p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively.Let v p be the normalized exponential valuation of C p with |p| p p −v p p 1/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 see 2, 5, 11 .From 3.1 , we can derive the following integral equation: The fermionic p-adic integral on Z p is defined by Kim as follows 2, 8, 9 : From 3.7 , we obtain the following integral equation: see 2 , where f n x f x n .Thus, by 3.8 , we have Let us take f y e t x y .Then we have From 3.10 , we have Thus, by 3.9 and 3.11 , we get

3.12
Let us consider the following p-adic integral on Z p :

3.13
From Theorem 2.1 and 3.13 , one has

3.14
Therefore, by 3.13 and 3.14 , we obtain the following theorem.

3.15
Now, one sets By Theorem 2.1, one gets

3.17
Therefore, by 3.16 and 3.17 , we obtain the following theorem.
Theorem 3.2.For n ∈ Z , one has Let us consider the following p-adic integral on Z p : From Theorem 2.2, one has

3.20
Therefore, by 3.19 and 3.20 , we obtain the following theorem.

Theorem 3 . 3 .Corollary 3 . 4 .G l 1
For n ∈ Z , one has 3.22 and 3.23 , we obtain the following corollary.For n ∈ Z , one has B n−l−k B k l 1 .3.24