In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Final

q-EULERNUMBERSANDPOLYNOMIALSASSOCIATEDWITHBASICZETAFUNCTIONS

arXiv:0807.2682v1 [math.NT] 17 Jul 2008

TaekyunKim

DivisionofGeneralEducation-Mathematics,KwangwoonUniversity,Seoul139-701,S.Korea

e-mail:tkkim@kw.ac.kr

Abstract.

Weconsidertheq-analogueofEulerzetafunctionwhichisde nedby

ζq,E(s)=[2]q

∞X( 1)nqnsn=1

Whenonetalksof

q-extension,qisvariouslyconsideredasanindeterminate,acomplexnumberq∈Corap-adicnumberq∈Cp.Ifq∈C,thenwenormallyassume|q|<1,andwhenq∈Cp,thenwenormallyassume|q 1|p<1.Weusethenotation:

[x]q=[x:q]=

1 qx

1+q

.

p.