q-Euler Numbers and Polynomials Associated with Basic Zeta F(4)
发布时间：2021-06-06
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
4q-ANALOGUEOFRIEMANNZETAFUNCTION
2.someidentitiesofq-EulernumbersEm,q
( m,1)
.
Inthissection,weassumeq∈Cpwith|1 q|p<1.By(1),weseethat
En,q
( n,1)
(x)=
q (n+1)t[x+t]ndµ q(t)
X
q=
[2]q
Thuswehave(5)
En,q( n,1)(x)=
[2]q
whered,narepositiveintegerswithd≡1(mod2).
Ifwetakex=0,thenwehave(6)
Em,q( m,1)
=
[2]q[2][n]mE( m,1)
qn
qm,qn
=
[2]q
k!
.
d
+t]nqddµ qd(t).
d
),
q-Euler Numbers and Polynomials Associated with Basic Zeta F(4).doc
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