On Genocchi Numbers and Polynomials

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rationalnumbers, the complex number field, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks about q-extension, q is variously considered as an indeterminate, a complex, q ∈ C, or a p-adic number, q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp, then we assume |q − 1|p < 1. The ordinary Genocchi polynomials are defined as the generating function:


Introduction
Let p be a fixed odd prime number. Throughout this paper, Z p , Q p , C, and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rationalnumbers, the complex number field, and the completion of the algebraic closure of Q p . Let v p be the normalized exponential valuation of C p with |p| p p −v p p 1/p. When one talks about q-extension, q is variously considered as an indeterminate, a complex, q ∈ C, or a p-adic number, q ∈ C p . If q ∈ C, one normally assumes |q| < 1. If q ∈ C p , then we assume |q − 1| p < 1. The ordinary Genocchi polynomials are defined as the generating function: For a fixed positive integer d with p, d 1, set a dp N Z p x ∈ X | x ≡ a mod dp N

2
Abstract and Applied Analysis cf. 1-30 , where a ∈ Z satisfies the condition 0 ≤ a < dp N . We say that f is uniformly differential function at a ∈ Z p and write f ∈ UD Z p if the difference quotients, F f x, y f x − f y / x − y , have a limit f a as x, y → a, a . Throughout this paper, we use the following notation: For f ∈ UD Z p , the fermionic p-adic invariant q-integral on Z p is defined as In this paper, we investigate some interesting integral equations related to I −1 f . From these integral equations related to I −1 f , we can derive many interesting properties of Genocchi numbers and polynomials. The main purpose of this paper is to derive the distribution relations of the Genocchi polynomials, and to constructthe Genocchi zeta function which interpolates the Genocchi polynomials at negative integers.

Genocchi numbers and polynomials
The Genocchi numbers are defined as where G n is replaced by G n , symbolically. The Genocchi polynomials are also defined as 2.3 Let f 1 x be translation with f 1 x f x 1 . Then we have the following integral equation.
Thus, we obtain For n ∈ N, we have By 2.6 and 2.7 , if we take f x x k k ∈ Z , we easily see that Thus, we have If n ≡ 1 mod 2 , then we know that Thus, we get

2.11
Let χ be the Dirichlet character with conductor d ∈ N, with d ≡ 1 mod 2 . Then, we consider the generalized Genocchi numbers attached to χ as follows:

Genocchi zeta function
Let F t, x be the generating function of G k x in complex plane as follows:

3.1
Then, we show that Therefore, we obtain the following proposition.
From Proposition 3.1, we can derive the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.
For s ∈ C, we define the Hurwitz-type Genocchi zeta function as follows.
Let χ be the Dirichlet character with conductor d ∈ N, with d ≡ 1 mod 2 , and let F χ t be the generating function in C of G n,χ . Then, we have From 3.7 , we derive d n t n n! .

3.11
Now, we consider the Dirichlet-type Genocchi -function in complex plane as follows. For s ∈ C, define G,χ s 2 ∞ n 1 −1 n χ n n s .

3.12
By 3.11 and 3.12 , we obtain the following theorem.
Theorem 3.4. Let χ be the Dirichlet character with conductor d ∈ N, with d ≡ 1 mod 2 , and let k ∈ Z . Then, one has G,χ 1 − k G k,χ k .