© Hindawi Publishing Corp. q-RIEMANN ZETA FUNCTION

We consider the modified q-analogue of Riemann zeta function
which is defined by
ζq(s)=∑n=1∞(qn(s−1)/[n]s), 0<q<1, s∈ℂ. In this paper, we give q-Bernoulli numbers
which can be viewed as interpolation of the above q-analogue of
Riemann zeta function at negative integers in the same way that
Riemann zeta function interpolates Bernoulli numbers at negative
integers. Also, we will treat some identities of q-Bernoulli
numbers using nonarchimedean q-integration.

1. Introduction.Throughout this paper, Z p , Q p , C, and C p will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Q p .
The p-adic absolute value in C p is normalized so that |p| p = 1/p.When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ C p .If q ∈ C, we normally assume |q| < 1.If q ∈ C p , then we normally assume |q − 1| p < p −1/(p−1) so that q x = exp(x log q) for |x| p ≤ 1.We use the notation Note that lim q→1 [x] = x for x ∈ Z p in the p-adic case.
Let UD(Z p ) be denoted by the set of uniformly differentiable functions on Z p .For f ∈ UD(Z p ), we start with the expression representing the analogue of Riemann's sums for f (cf.[4]).
The integral of f on Z p will be defined as the limit (N → ∞) of these sums, which exists.The p-adic q-integral of a function f ∈ UD(Z p ) is defined by (see [4]) For d that is a fixed positive integer with (p, d) = 1, let where a ∈ Z lies in 0 ≤ a < dp N .Let N be the set of positive integers.For m, k ∈ N, the q-Bernoulli polynomials, β (−m,k) m (x, q), of higher order for the variable x in C p are defined using p-adic q-integral by (cf.[4]) (1.5) Now, we define the q-Bernoulli numbers of higher order as follows (cf.[2,4,7]): ( By (1.5), it is known that (cf.[4]) where m i are the binomial coefficients.
Note that lim q→1 β m are ordinary Bernoulli numbers of order k (cf.[2,3,5,7,9]).By (1.5) and (1.7), it is easy to see that (1.8) We modify the q-analogue of Riemann zeta function which is defined in [1] as follows: for q ∈ C with 0 < q < 1, s ∈ C, define (1.9) The numerator ensures the analytic continuation for (s) > 1.In (1.9), we can consider the following problem."Are there q-Bernoulli numbers which can be viewed as interpolation of ζ q (s) at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers?" In this paper, we give the value ζ q (−m) for m ∈ N, which is the answer of the above problem, and construct a new complex q-analogue of Hurwitz's zeta function and q-Lseries.Also, we will treat some interesting identities of q-Bernoulli numbers.
If we take x = 0, then we have 3) It is easy to see that lim q→1 β (−m,1) m = B m , where B m are ordinary Bernoulli numbers (cf.[7]).
Remark 2.1.By (2.3), note that Let F q (t) be the generating function of β (−n,1) n as follows: By (1.7) and (2.5), we easily see that Through differentiating both sides with respect to t in (2.5) and (2.6), and comparing coefficients, we obtain the following proposition.
Let χ be a primitive Dirichlet character with conductor f ∈ N.For m ∈ N, we define (2.8) Note that (2.9) 3. q-analogs of zeta functions.In this section, we assume q ∈ C with |q| < 1.In [1], the q-analogue of Riemann zeta function was defined by (cf.[1]) Now, we modify the above q-analogue of Riemann zeta function as follows: for q ∈ C with 0 By (2.5), (2.6), and (2.7), we obtain the following proposition.
(3.3) Hence, we can define q-analogue of Hurwitz ζ-function as follows: for s ∈ C, define [n]q x + [x] s . (3.4) Note that ζ q (s, x) has an analytic continuation in C with only one simple pole at s = 1.By (3.3) and (3.4), we have the following theorem.
Theorem 3.2.For any positive integer k, there exists Let χ be Dirichlet character with conductor d ∈ N. By (2.9), the generalized q-Bernoulli numbers with χ can be defined by For s ∈ C, we define It is easy to see that Let a and F be integers with 0 < a < F. For s ∈ C, we consider the functions H q (s,a,F) as follows: Then we have where n is any positive integer.
Therefore, we obtain the following theorem.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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