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FunctionesetApproximatio
51.1(2014),7–22
doi:10.7169/facm/2014.51.1.1
POLY-EULERPOLYNOMIALSANDARAKAWA–KANEKOTYPE
ZETAFUNCTIONS
YoshinoriHamahata
Abstract:Weintroducepoly-Eulerpolynomials,whichgeneralizeEulerpolynomials.Various
resultsaboutthemareprovided.Furthermore,weintroducezetafunctionsofArakawa–Kaneko
type,anddiscusstheirpropertiesandtherelationwithpoly-Eulerpolynomials.
Keywords:polylogarithms,Eulernumbersandpolynomials,Bernoullinumbersandpolynomi-
als,zetafunction.
1.Introduction
EulerpolynomialsEn(x)(n=0j1j2j...)aredefinedbythegeneratingfunction
et+1
2ext
=
n=o
Σ
∞
En(x)
tn
n!
.
ThefirstfewvaluesareEo(x)=1,E1(x)=x−1/2,E2(x)=x2−x,E3(x)=
x3−3x2/2+1/4.Letx>0.WedefinetheEulerzetafunctionofHurwitztypeby
CE(sjx)=2
n=o
Σ
∞
(n+x)5
(−1)n
(1.1)
forRe(s)>0.Thisfunctionisanalyticallycontinuedtothewholecomplexs-
planeasanentirefunction.Infact,thisfollowsfromthefactthatforA∈R\Z,
theLerchzetafunction
L(Ajxjs)=
n=o
Σ
∞
(n+x)5
e2πżAn
j
Re(s)>0
isanalyticallycontinuedtothewholecomplexs-planeasanentirefunction
(see[13,(2.2)]).Itisknownthatforeachnon-negativeintegern,CE(−njx)=
En(x).
2010MathematicsSubjectClassification:primary:11B68;secondary:11M32,11M35
