A note on the -zeta-type function with weight α
© Cetin et al.; licensee Springer 2013
Received: 28 January 2013
Accepted: 20 February 2013
Published: 13 March 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.
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
By considering the above identity in (2.3), we can easily state the following theorem.
Putting in (2.4) leads to the following corollary.
Taking into Theorem 1, we have the symmetric property of -Genocchi polynomials by the following theorem.
Thus, by comparing the coefficients of , we state the following corollary.
By using Theorem 2 and (2.5), we readily derive the following symmetric relation after some applications.
When into Theorem 3, it leads to the following corollary.
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.
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