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

2q-ANALOGUEOFRIEMANNZETAFUNCTION

Notethatlimq→1[x]q=xforx∈Zpinpresentedp-adiccase.

LetUD(Zp)bedenotedbythesetofuniformlydi erentiablefunctionsonZp.Forf∈UD(Zp),letusstartwiththeexpression

1

[pN] q

0≤j<pN

f(j)( q)j,(see[5,6,16]).

Forda xedpositiveintegerwith(p,d)=1,let

N

X=Xd=←lim Z/dpZ,X1=Zp,

N

X=

0<a<dp

(a,p)=1

a+dpZp,

a+dpNZp={x∈X|x≡a(moddpN)},

wherea∈Zliesin0≤a<dpN,(see[1-30]).

LetNbethesetofpositiveintegers.Form,k∈N,theq-Eulerpolynomials( m,k)Em(x,q)ofhigherorderinthevariablesxinCpbymakinguseofthep-adicq-integral,cf.[5,6],arede nedby

( m,k)

(1)Em,q(x)=

Zp

···

Zp

Zp

[x+x1+x2+···+xk]mq

·q

ktimes

x1(m+1) x2(m+2) ··· xk(m+k)

dµ q(x1)dµ q(x2)···dµ q(xk).

Now,wede netheq-Eulernumbersofhigherorderasfollows:

( m,k)( m,k)Em,q=Em,q(0).