New Construction Weighted h , q-Genocchi Numbers and Polynomials Related to Zeta Type Functions

The fundamental aim of this paper is to construct h, q -Genocchi numbers and polynomials with weight α. We shall obtain some interesting relations by using p-adic q-integral on Zp in the sense of fermionic. Also, we shall derive the h, q -extensions of zeta type functions with weight α from the Mellin transformation of this generating function which interpolates the h, q -Genocchi numbers and polynomials with weight α at negative integers.


Introduction, Definitions, and Notations
Let p be a fixed odd prime number.Throughout this paper we use the following notations.Z p denotes the ring of p-adic rational integers, Q denotes the field of rational numbers, Q p denotes the field of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p .Let N be the set of natural numbers and N * N ∪ {0}.The p-adic absolute value is defined by |p| p 1/p.In this paper, we assume |q − 1| p < 1 as an indeterminate.In 1-3 , Kim defined the fermionic p-adic q-integral on Z p as follows: x q is a q-extension of x which is defined by Note that lim q → 1 x q x.Let f n x f x n .By the definition 1.1 we easily get Continuing this process, we obtain easily the relation h, q -Genocchi numbers are defined as follows: with the usual convention about replacing G h q n by G h n,q see 6 .In this paper, we constructed h, q -Genocchi numbers and polynomials with weight α.By using fermionic p-adic q-integral equations on Z p , we investigated some interesting identities and relations on the h, q -Genocchi numbers and polynomials with weight α.Furthermore, we derive the q-extensions of zeta type functions with weight α from the Mellin transformation of this generating function which interpolates the h, q -Genocchi polynomials with weight α.

On the Weighted h, q -Genocchi Numbers and Polynomials
In this section, by using fermionic p-adic q-integral equations on Z p , some interesting identities and relation on the h, q -Genocchi numbers and polynomials with weight α are shown.
Definition 2.1.Let α, n ∈ N * and h ∈ N. Then the h, q -Genocchi numbers with weight α defined by as follows: If we take h 1 to 2.1 , then we have, Therefore, we obtain the following theorem.

2.4
From 12 , we obtain h, q -Genocchi numbers with weight α witt's type formula as follows.

Theorem 2.3. For α, n ∈ N
From 2.1 , one easily gets −1 m q mh e t m q α .2.6 By 2.6 , one has −1 m q mh e t m q α .2.7 Therefore, we obtain the following corollary.
Now, one considers the h, q -Genocchi polynomials with weight α as follows: From 2.9 , one sees that x t n /n! .Then, one has By 1.4 , one sees that Therefore, we obtain the following theorem.
Theorem 2.5.For m, h ∈ N, and α, n ∈ N * , one has In 1.3 , it is known that If we take f x q h−1 x e t x q α , then one has

2.15
Therefore, by 2.15 , we obtain the following theorem.

2.16
From 2.9 , one can easily derive the following:

2.17
Therefore, by 2.17 , we obtain the following theorem.

Interpolation Function of the Polynomials G α,h n,q x
In this section, we give interpolation function of the generating functions of h, q -Genocchi polynomials with weight α.For s ∈ C and h ∈ N, by applying the Mellin transformation to 2.11 , we obtain x q α dt, 3.1 so we have

3.2
We define q-extension zeta type function as follows.

I
α,h q s, x can be continued analytically to an entire function.
By subsituting s −n into 3.3 one easily gets I α,h q −n, x G α,h n 1,q x n 1 .

3.4
We obtain the following theorem.