Some Properties of Multiple Generalized q-Genocchi Polynomials with Weight α and Weak Weight β

The present paper deals with the various q-Genocchi numbers and polynomials. We define a new type ofmultiple generalized q-Genocchi numbers and polynomials withweight α andweakweight β by applying the method of p-adic q-integral. We will find a link between their numbers and polynomials with weight α and weak weight β. Also we will obtain the interesting properties of their numbers and polynomials with weight α and weak weight β. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalized q-Genocchi polynomials with weight α and weak weight β and find some combinatorial relations.


Introduction
Let p be a fixed odd prime number.Throughout this paper Z p , Q p , C, and C p denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Q p , respectively.Let N be the set of natural numbers and Z N ∪ {0}.Let v p be the normalized exponential valuation of C p with |p| p p −v p p 1/p see 1-21 .When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ C p .If q ∈ C, then one normally assumes |q| < 1.If q ∈ C p , then we assume that |q − 1| p < 1.
Throughout this paper, we use the following notation: Hence lim q → 1 x q x for all x ∈ Z p see 1-14, 16, 18, 20, 21 .

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We say that g : Z p → C p is uniformly differentiable function at a point a ∈ Z p and we write g ∈ UD Z p if the difference quotients Φ g : Z p × Z p → C p such that Φ g x, y g x − g y x − y 1.2 have a limit g a as x, y → a, a .
Let d be a fixed integer, and let p be a fixed prime number.For any positive integer N, we set where a ∈ Z lies in 0 ≤ a < dp N .For any positive integer N, μ q a dp N Z p q a dp N q 1.4 is known to be a distribution on X.
For g ∈ UD Z p , Kim defined the q-deformed fermionic p-adic integral on Z p : x .1.5 see 1-13 , and note that We consider the case q ∈ −1, 0 corresponding to q-deformed fermionic certain and annihilation operators and the literature given there in 9, 13, 14 .In 9, 12, 14, 19 , we introduced multiple generalized Genocchi number and polynomials.Let χ be a primitive Dirichlet character of conductor f ∈ N. We assume that f is odd.Then the multiple generalized Genocchi numbers, G r n,χ , and the multiple generalized Genocchi polynomials, G r n,χ x , associated with χ, are defined by

1.7
In the special case x 0, G r n,χ G r n,χ 0 are called the nth multiple generalized Genocchi numbers attached to χ.Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to their q-analogues.In generalizing the generating functions of the Genocchi numbers and polynomials to their respective qanalogues; it is more useful than defining the generating function for the Genocchi numbers and polynomials see 12 .
Our aim in this paper is to define multiple generalized q-Genocchi numbers G α,β,r n,χ,q and polynomials G α,β,r n,χ,q x with weight α and weak weight β.We investigate some properties which are related to multiple generalized q-Genocchi numbers G α,β,r n,χ,q and polynomials G α,β,r n,χ,q x with weight α and weak weight β.We also derive the existence of a specific interpolation function which interpolate multiple generalized q-Genocchi numbers G α,β,r n,χ,q and polynomials G α,β,r n,χ,q x with weight α and weak weight β at negative integers.

The Generating Functions of Multiple Generalized q-Genocchi
Numbers and Polynomials with Weight α and Weak Weight β Many mathematicians constructed various kinds of generating functions of the q-Gnocchi numbers and polynomials by using p-adic q-Vokenborn integral.First we introduce multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β.Let us define the generalized q-Genocchi numbers G α,β n,χ,q and polynomials G α,β n,χ,q x with weight α and weak weight β, respectively, tχ y e x y q α t dμ −q β y .

2.1
By using the Taylor expansion of e x q α t , we have

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By comparing the coefficient of both sides of t n /n! in 2.2 , we get From 2.2 and 2.3 , we can easily obtain that −1 l q βl χ l e l q α t .

2.4
Therefore, we obtain Similarly, we find the generating function of generalized q-Genocchi polynomials with weight α and weak weight β:

2.6
From 2.6 , we have First, we define the multiple generalized q-Genocchi numbers G α,β,r n,χ,q with weight α and weak weight β: t n n! .

2.8
Then we have where n r r n r !/n!r!.By comparing the coefficients on the both sides of 2.9 , we obtain the following theorem.

2.10
From now on, we define the multiple generalized q-Genocchi polynomials G α,β,r n,χ,q x with weight α and weak weight β.
t n n! .

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Then we have where n r r n r !/n!r!.By comparing the coefficients on the both sides of 2.12 , we have the following theorem.

2.13
In 2.11 , we simply identify that lim

2.14
So far, we have studied the generating functions of the multiple generalized q-Genocchi numbers G α,β,r n,χ,q and polynomials G α,β,r n,χ,q x with weight α and weak weight β.
International Journal of Mathematics and Mathematical Sciences 7

Modified Multiple Generalized q-Genocchi Polynomials with Weight α and Weak Weight β
In this section, we will investigate about modified multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β.Also, we will find their relations in multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β.
Firstly, we modify generating functions of G α,β,r n,χ,q and G α,β,r n,χ,q x .We access some relations connected to these numbers and polynomials with weight α and weak weight β.For this reason, we assign generating function of modified multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β which are implied by G α,β,r n,χ,q and G α,β,r n,χ,q x .We give relations between these numbers and polynomials with weight α and weak weight β.
We modify 2.11 as follows: where F α,β,r χ,q t, x is defined in 2.11 .From the above we know that After some elementary calculations, we attain F α,β,r χ,q t, x q −αrx e q −αx x q α t F α,β,r χ,q t , 3.3 where F α,β,r χ,q t is defined in 2.8 .From the above, we can assign the modified multiple generalized q-Genocchi polynomials ε α,β,r n,χ,q x with weight α and weak weight β as follows: Then we have International Journal of Mathematics and Mathematical Sciences Corollary 3.2.For r ∈ N and n ∈ Z , by using 3.7 , one easily obtains Secandly, by using generating function of the multiple generalized q-Genocchi polynomials with weight α and weak weight β, which is defined by 2.11 , we obtain the following identities.
By using 2.13 , we find that 1 − q α l 1 q f{α a n−l β} r .

3.8
Thus we have the following theorem.

3.9
By using 2.13 , we have

3.10
International Journal of Mathematics and Mathematical Sciences 9 Thus we have

3.11
By comparing the coefficients of both sides of n r !/t n r in the above, we arrive at the following theorem.

3.12
From 2.12 , we easily know that t n r n r ! .

3.13
From the above, we get the following theorem.

3.14
International Journal of Mathematics and Mathematical Sciences From 2.13 , we have

3.15
By using Cauchy product in 3.15 , we obtain r i 1 a i fj x q α t e s i 1 b i f n−j x q α t .

3.16
From 3.16 , we have

3.17
By comparing the coefficients of both sides of t m r s / m r s ! in 3.17 , we have the following theorem.

3.19
By using 2.13 we have the following theorem.

Interpolation Function of Multiple Generalized q-Genocchi Polynomials with Weight α and Weak Weight β
In this section, we see interpolation function of multiple generalized q-Genocchi polynomials with weak weight α and find some relations.

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Let us define interpolation function of the G α,β,r k r,q x as follows.

Definition 4 . 1 .Setting 3 .Theorem 4 . 3 .
Let q, s ∈ C with |q| < 1 and 0 < x ≤ 1.Then one defines the multiple generalized Hurwitz type q-zeta funtion.In 4.1 , setting r 1, we have Substituting s −n, n ∈ Z into 4.1 , then we have, 14 into the above, we easily get the following theorem.Let r ∈ N, n ∈ Z .Then one has