q-Bernoulli Numbers and Polynomials Associated with Multiple q-Zeta Functions and Basic L-series

H. M. Srivastava*, T. Kim**, and Y. Simsek***

*Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada
**Institute of Science Education, Kongju National University, Kongju 314-701, South Korea
***Department of Mathematics, Faculty of Science, Mersin University, TR-33343 Mersin, Turkey
harimsri@math.uvic.ca | tkim@mail.kongju.ac.kr | ysimsek@mersin.edu.tr

Received January 5, 2005

Abstract. The main purpose of this paper is to present a systematic study of some families of multiple q-zeta functions and basic (or q-) L-series. In particular, by using the q-Volkenborn integration and uniform differentiation on Z_p, we construct p-adic q-zeta functions. These functions interpolate the q-Bernoulli numbers and polynomials. The values of p-adic q-zeta functions at negative integers are given explicitly. We also define new generating functions of q-Bernoulli numbers and polynomials. By using these functions, we prove the analytic continuation of some basic (or q-) L-series. These generating functions also interpolate Barnes' type Changhee q-Bernoulli numbers with attached Dirichlet character. By applying the Mellin transformation, we obtain relations between Barnes' type q-zeta function and new Barnes' type Changhee q-Bernoulli numbers. Furthermore, we construct the Dirichlet type Changhee basic (or q-) L-functions.

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