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18
YoshinoriHamahata
Proof.Theleft-handsideoftheidentityinthetheoremequals
Γ(s)/
2
o
∞
Lik(1−elt)
t(1+et)
t(1+et)elxtt5l1dt
=
Γ(s)/
2
o
∞
Lik(1−elt)
1−elt
(1−elt)elxtt5l1dj
whichisnothingbuttheright-handoftheidentity.
.
5.Generalizedpoly-EulerpolynomialsandArakawa–Kanekotype
L-functions
Inthissection,usingaDirichletcharacter,weextendpoly-Eulerpolynomialsand
Arakawa–Kanekotypezetafunctions.
5.1.Generalizedpoly-Eulerpolynomials
LetfbeapositiveintegerandχtheDirichletcharacterwithconductorf=fχ.As
iswell-known,generalizedEulerpolynomialsaredefinedbythegeneratingfunction
2
fl1
Σ
a=o
(−1)aχ(a)
eft+1
eat
ext=
n=o
Σ
∞
En,χ(x)
tn
n!
.
Definition5.1.Letk∈Z.Wedefinegeneralizedpoly-EulerpolynomialsE
n,χ(x)
(k)
(n=0j1j2j...)by
f
2
fl1
Σ
a=o
(−1)aχ(a)
Lik(1−elft)
t(eft+1)
e(x+a)t=
n=o
Σ
∞
E(k)
n,χ(x)
tn
n!
.
WecallE
n,χ:=E
(k)
n,χ(0)(n=0j1j2j...)generalizedpoly-Eulernumbers.
(k)
Onecaneasilyprovethefollowingtwotheoremsasinpoly-Eulerpolynomial
case.
Theorem5.2(Additionformula).Fork∈Z,n>0,wehave
E(k)
n,χ(x+y)=
m=o
Σ
n
(n
m)E(k)
m,χ(x)ynlm.
Theorem5.3(Appellsequence).Fork∈Z,n>0,wehave
dx
d
E
n+1,χ(x)=(n+1)E(k)
(k)
n,χ(x).
WehavethefollowingexpressionsofE
n,χ(x)intermsofE
(k)
n(x)andEn(x).
(k)
