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Poly-EulerpolynomialsandArakawa–Kanekotypezetafunctions
19
Theorem5.4.Foranyn>0,wehave
E(k)
n,χ(x)=fn
Σ
a=o
fl1
(−1)aχ(a)E(k)
n
(x+a
f
)j
E(k)
n,χ=fn
Σ
a=o
fl1
(−1)aχ(a)E(k)
n
(a
f).
Theorem5.5.Fork∈Z,n>0,wehave
E(k)
n,χ(x)
=
n+1
fn
fl1
Σ
a=o
(−1)aχ(a)
m=o
Σ
∞
(m+1)k
1
m+1
Σ
j=o
(−1)j(m+1
j
)En+1(x+a
f
−fj
).
5.2.Arakawa–KanekotypeL-functions
Definition5.6.Fork∈Z,definetheL-seriesattachedtoχbytheLaplace–Mellin
integral
LE,k(sjxjχ)=
f
2
fl1
Σ
a=o
(−1)aχ(a)
Γ(s)/
1
o
∞
Lik(1−elft)
t(eft+1)
e
l(xla)tt5l1dt.
ByProposition4.2,LE,k(sjxjχ)isdefinedforRe(s)>1andx>0ifk>1,
andRe(s)>1andx>|k|+1ifk<0.WecallLE,k(sjxjχ)theArakawa–
KanekotypeL-function.ThesefunctionsincludeArakawa–KanekoandHurwitz
L-functions:
Theorem5.7.Onehas
LE,k(sjxjχ)=f
l5
Σ
a=o
fl1
(−1)aχ(a)ZE,k(sjx
−a
f
).
Especially,
LE,1(sjxjχ)=f
l5
fl1
Σ
a=o
(−1)aχ(a)CE(sjx
−a
f
+1).
Furthermore,ifχ=χo,thetrivialcharacter,then
LE,1(sjxjχo)=ZE,1(sjx)=CE(sjx+1).
(5.1)
Theorem5.8.Thefunctionsl→LE,k(sjxjχ)hasanalyticcontinuationtoan
entirefunctiononthewholecomplexs-planeandhastheidentity
LE,k(−njxjχ)=(−1)nE(k)
n,χ(−x)
forn>0,x>0.
