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A note on the -zeta-type function with weight α
Journal of Inequalities and Applications volume 2013, Article number: 100 (2013)
The objective of this paper is to derive the symmetric property of an -zeta function with weight α. By using this property, we give some interesting identities for -Genocchi polynomials with weight α. As a result, our applications possess a number of interesting properties which we state in this paper.
Recently, Kim has developed a new method by using the q-Volkenborn integral (or p-adic q-integral on ) and has added weight to q-Bernoulli numbers and polynomials and investigated their interesting properties (see ). He also showed that these polynomials are closely related to weighted q-Bernstein polynomials and derived novel properties of q-Bernoulli numbers with weight α by using the symmetric property of weighted q-Bernstein polynomials with the help of the q-Volkenborn integral (for more details, see ). Afterward, Araci et al. have introduced weighted -Genocchi polynomials and defined -zeta-type function with weight α by applying the Mellin transformation to the generating function of the -Genocchi polynomials with weight α which interpolates for -Genocchi polynomials with weight α at negative integers (for details, see ). In this paper, we also consider a -zeta-type function with weight α and derive some interesting properties.
We firstly list some notations as follows.
Imagine that p is a fixed odd prime. Throughout this work, ℤ, , and will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of , respectively. Also, we denote and . Let (ℚ is the field of rational numbers) denote the p-adic valuation of normalized so that . The absolute value on will be denoted as , and for . When one speaks of q-extensions, q is considered in many ways, e.g., as an indeterminate, a complex number , or a p-adic number . If , we assume that . If , we assume so that for . We use the following notation:
For a fixed positive integer d, set
The following p-adic q-Haar distribution was defined by Kim:
Let be the set of uniformly differentiable functions on . We say that f is a uniformly differentiable function at a point if the difference quotient
The bosonic integral is considered as the bosonic limit , . Similarly, the p-adic fermionic integration on is defined by Kim  as follows:
By using a fermionic p-adic q-integral on , -Genocchi polynomials are defined by 
For in (1.4), we have are called -Genocchi numbers with weight α which is defined by
By (1.4), we have a distribution formula for -Genocchi polynomials, which is shown by 
By applying some elementary methods, we will give symmetric properties of weighted -Genocchi polynomials and a weighted -zeta-type function. Consequently, our applications seem to be interesting and worthwhile for further works of many mathematicians in analytic numbers theory.
2 On the -zeta-type function
In this part, we firstly recall the -zeta-type function with weight α which is derived in  as follows:
where , and . It is clear that the special case and in (2.1) reduces to the ordinary Hurwitz-Euler zeta function. Now, we consider (2.1) in the following 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 with . For and . Thus, we can compute as follows:
From this, we can easily discover the following:
Replacing a by b and j by i in (2.2), we have the following:
By considering the above identity in (2.3), we can easily state the following theorem.
Theorem 1 The following identity is true:
Now, setting in Theorem 1, we have the following distribution formula:
Putting in (2.4) leads to the following corollary.
Corollary 1 The following identity holds true:
Taking into Theorem 1, we have the symmetric property of -Genocchi polynomials by the following theorem.
Theorem 2 The following identity is true:
Now also, setting and replacing x by in the above theorem, we can rewrite the following -Genocchi polynomials with weight α:
Due to Araci et al. , we develop as follows:
By using the Cauchy product, we see that
Thus, by comparing the coefficients of , we state the following corollary.
Corollary 2 The following equality holds true:
By using Theorem 2 and (2.5), we readily derive the following symmetric relation after some applications.
Theorem 3 The following equality holds true:
When into Theorem 3, it leads to the following corollary.
Corollary 3 The following identity holds true:
where and are called the ordinary Genocchi polynomials which are defined via the following generating function:
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Dedicated to Professor Hari M Srivastava.
We would like to thank the referees for their valuable suggestions and comments on the present paper. The third author is supported by Uludag University Research Fund, Project Number F-2012/16 and F-2012/19.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Cetin, E., Acikgoz, M., Cangul, I.N. et al. A note on the -zeta-type function with weight α. J Inequal Appl 2013, 100 (2013). https://doi.org/10.1186/1029-242X-2013-100
- -Genocchi numbers and polynomials with weight α
- -zeta function with weight α
- p-adic q-integral on