EXPLORING THE q-RIEMANN ZETA FUNCTION AND q-BERNOULLI POLYNOMIALS

We study that the q-Bernoulli polynomials, which were constructed by Kim, are analytic continued to βs(z). A new formula for the q-Riemann zeta function ζq(s) due to Kim in terms of nested series of ζq(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenomenon of “scattering” of the zeros of βs(z) is observed. Following the idea of q-zeta function due to Kim, we are going to use “Mathematica” to explore a formula for ζq(n).


Introduction
Throughout this paper, Z, R, and C will denote the ring of integers, the field of real numbers, and the complex numbers, respectively.
When one talks of q-extension, q is variously considered as an indeterminate, a complex number, or a p-adic number.In the complex number field, we will assume that |q| < 1 or |q| > 1.The q-symbol [x] q denotes [x] q = (1 − q x )/(1 − q).
In this paper, we study that the q-Bernoulli polynomials due to Kim (see [2,8]) are analytic continued to β s (z).By those results, we give a new formula for the q-Riemann zeta function due to Kim (cf.[4,6,8]) and investigate the new concept of dynamics of the zeros of analytic continued polynomials.Also, we observe an interesting phenomenon of "scattering" of the zeros of β s (z).Finally, we are going to use a software package called "Mathematica" to explore dynamics of the zeros from analytic continuation for q-zeta function due to Kim.
By (2.1), we easily see that where n j is a binomial coefficient.In (2.1), it is easy to see that with the usual convention of replacing β n (h | q) by β n (h | q).By differentiating both sides with respect to t in (2.1), we have q hn [n] m q .

Analytic continuation of q-Bernoulli polynomials
For consistency with the redefinition of β n = β(n) in (4.5) and (4.6), (5.1) The analytic continuation can be then obtained as where [s] gives the integer part of s, and so s − [s] gives the fractional part.Deformation of the curve β(2,w) into the curve β(3,w) via the real analytic continuation β(s,w), 2 ≤ s ≤ 3, −0.5 ≤ w ≤ 0.5.