T. Kim, C. S. Ryoo, L. C. Jang and S. H. Rim

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 11, 2005, Number 1, Pages 7–19

**Full paper (PDF, 334 Kb)**

## Details

### Authors and affiliations

T. Kim

*Institute of Science Education
Kongju National University, Kongju 314-701, Korea*

C. S. Ryoo

*Department of Mathematics,
Hannam University, Daejeon 306-791, Korea*

L. C. Jang

*Department of Mathematics and Compute Science,
KonKuk University, Choongju 380-701, Korea*

S. H. Rim

*Department of Mathematics Education,
Kyungpook University, Daegu 702-701, Korea*

### Abstract

In this paper we study that the q-Bernoulli polynomial, which were constructed by T.Kim, are analytic continued to β_{s}(z). A new formula for the q-Riemann Zeta function ζ_{,}(s) due to T.Kim (see [1,2,8]) in terms of nested series of ζ_{,}(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an investing phenomenon of ‘scattering’ of the zeros of β_{s}(z) is observed. Following the idea of q-zeta function due to T.Kim, we are going to use “Mathematica” to explore a formula for ζ_{,}(n).

### Keywords

- q-Bernoulli polynomial
- q-Riemann Zeta function

### AMS Classification

- 11B68
- 11S40

### References

- T. Kim, S. H. Rim, ‘Generalized Carlitz’s q-Bernoulli Numbers in the

p-adic number field ’, Adv. Stud. Contemp. Math., 2, 9-19 (2000). - T. Kim, ‘q-Volkenborn integration ’, Russ. J. Math. Phys., 9, 288-299

(2002). - T. Kim, ‘Non-Arichimedean q-integrals associated with multiple

Changhee q-Bernoulli polynomials ’, Russ. J. Math. Phys., 10, 91-98

(2003). - T. Kim, ‘Analytic continuation of multiple q-Zeta functions and their

values at negative integers ’, to appear in Russ. J. Math. Phys., 11 (2),

(2004). - T. Kim, ‘A note on multiple zeta functions’, JP J. Algebra, Number

Theory and Application, 3 (3), 471-476 (2003). - T. Kim, ‘A note on Dirichlet L-series’, Proc. Jangjeon Math. Soc., 6

(2), 161-166 (2003). - T. Kim, ‘ q-Riemann zeta functions ’, to appear in Int. J. Math. Math.

Sci., (2004). - T. Kim, ‘On p-adic q-L-function and sums of powers ’, Discrete Math.,

252, 179-187 (2002).

## Related papers

## Cite this paper

Kim, T., Ryoo, C. S., Jang, L. C., and Rim, S. K. (2005). Exploring the *q*-Riemann Zeta function and *q*-Bernoulli polynomials. *Notes on Number Theory and Discrete Mathematics*, 11(1), 7-19.