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

3

From(1),wecanderive

( m,k)Em,q

(2)

=lim

1

m m

( 1)iii=0

N→∞

1

(1 q)m

et+1

k

=

n=0

∞

(k)tEn

n

(1 q)m

m

m

( 1)jqjx

jj=0

1

[n]sq

,q∈Rwith0<q<1ands∈C.

Thenumeratorensurestheconvergence.In(4),wecanconsiderthefollowingproblem:

“Arethereq-Eulernumberswhichcanbeviewedasinterpolatingofζq,E(s)atnegativeintegers,inthesamewaythatRiemannzetafunctioninterpolatesBernoullinumbersatnegativeintegers”?

Inthispaper,wegivethevalueζq,E( m),form∈N,whichisaansweroftheaboveproblemandconstructanewcomplexq-analogueofHurwitz’stypeEulerzetafunctionandq-L-seriesrelatedtoq-Eulernumbers.Also,wewilltreatsomeinterestingidentitiesofq-Eulernumbers.