A note on the (h,q)-Zeta type function with weight alpha

The objective of this paper is to derive symmetric property of (h,q)-Zeta function with weight alpha. By using this property, we give some interesting identities for (h,q)-Genocchi polynomials with weight alpha. As a result, our applications possess a number of interesting property which we state in this paper.


INTRODUCTION
Recently, T. Kim has developed a new method by using q-Volkenborn integral (or p-adic q-integral on Z p ) which has added a weight to q-Bernoulli polynomials and investigated their properties (see [8]). He also showed that this polynomials are closely related to weighted q-Bernstein polynomials and derived novel properties of q-Bernoulli numbers with weight α by using symmetric property of weighted q-Bernstein polynomials on the q-Volkenborn integral (for more details, see [10]). After, Araci et al. have introduced weighted (h, q)-Genocchi polynomials and so defined (h, q)-Zeta type function with weight by applying Mellin transformation to generating function of (h, q)-Genocchi polynomials with weight α which interpolates for (h, q)-Genocchi polynomials with weight α at negative integers (for details, see [20]). In this paper, we also consider (h,q)-Zeta type function with weight and derive some interesting properties.
We firstly list some notations as follows: Imagine that p be a fixed odd prime. Throughout this work Z, Z p , Q p and C p will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p , respectively. Also we denote N * = N ∪ {0} and exp (x) = e x . Let v p : C p → Q ∪ {∞} (Q is the field of rational numbers) denote the p-adic valuation of C p normalized so that v p (p) = 1. The absolute value on C p will be denoted as | .|, and |x| p = p −vp(x) for x ∈ C p . When one speaks of q-extensions, q is considered in many ways, e.g. as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ C p , If q ∈ C we assume that |q| < 1. If q ∈ C p , we assume |1 − q| p < p − 1 p−1 , so that q x = exp (x log q) for |x| p ≤ 1. We use the following where we want to note that lim q→1 [x] q = x; cf. . For a fixed positive integer d, set a + dpZ p and a + dp n Z p = {x ∈ X | x ≡ a (mod dp n )} , where a ∈ Z satisfies the condition 0 ≤ a < dp n (see ).
The following q-Haar distribution is defined by T. Kim for any positive n (see [11], [12]). Let U D (Z p ) be the set of uniformly differentiable function on Z p . We say that f is a uniformly differentiable function at a point a ∈ Z p , if the difference quotient x − y has a limit f´(a) as (x, y) → (a, a) and denote this by f ∈ U D (Z p ) . In [11] and [12], the p-adic q-integral of the function f ∈ U D (Z p ) is defined by Kim The bosonic integral is considered as the bosonic limit q → 1, I 1 (f ) = lim q→1 I q (f ). Similarly, the p-adic fermionic integration on Z p is defined by Kim [5] as follows: By using fermionic p-adic q-integral on Z p , (h, q)-Genocchi polynomials are defined by [20] are called (h, q)-Genocchi numbers with weight α which is defined by By (4), we have distribution formula for (h, q)-Genocchi polynomials, which is shown by [20] By applying some elementary methods, we shall give symmetric properties of weighted (h, q)-Genocchi polynomials and weighted (h, q)-Zeta type function. Consequently, our applications seem to be interesting and worthwhile for studying in Theory of Analytic Numbers.

ON THE (h, q)-ZETA-TYPE FUNCTION
In this part, we firstly recall the (h, q)-Zeta type function with weight α which is derived in [20] as follows: where q ∈ C, h ∈ N and ℜ (s) > 1. It is clear that the special case h = 0 and q → 1 in (5), it reduces to the ordinary Hurwitz-Euler zeta function. Now, we consider (5) in this form By applying some basic operations to the above identity, that is, for any positive integers m and b, there exist unique non-negative integers k and i such that m = bk +i with 0 ≤ i ≤ b−1. For a ≡ 1(mod 2) and b ≡ 1(mod 2). Thus, we can compute as follows: From this, we can easily discover the following Replacing a by b and j by i in (6) and so we have the following By considering the above identity in (7), we can easily state the following theorem.
Theorem 2.1. The following Now, setting b = 1 in Theorem 2.1, we have the following distribution formula If putting a = 2 in (8) leads to the following corollary.
Corollary 2.2. The following identity holds true: Taking s = −m into Theorem 2.1, we have the symmetric property of (h, q)-Genocchi polynomials by the following theorem.
Theorem 2.3. The following identity is true.
Now also, setting b = 1 and replacing x by x a on the above theorem, we can rewrite the following (h, q)-Genocchi polynomials with weight α.
Due to Araci et al. [20], we develop as follows t n n! by using Cauchy product, we see that Thus, by comparing the coefficients of t n n! , we state the following corollary. Corollary 2.4. The following equality holds true: By using Theorem 2.3 and (9), we readily derive the following symmetric relation after some applications.
Theorem 2.5. The following equality holds true: i,q a (bx) S  where S m (a) = a−1 j=0 (−1) j j m and G n (x) are called the ordinary Genocchi polynomials which is defined via the following generating function ∞ n=0 G n (x) t n n! = 2t e t + 1 e xt .