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8
YoshinoriHamahata
Inthisarticle,weintroducepoly-Eulerpolynomials,whichgeneralizeEuler
polynomials.Thesepoly-Eulerpolynomialsaredifferentfromthosedefinedin
Son-Kim[15].Variousresultsaboutthemareprovided.Furthermore,weintro-
ducezetafunctionsofArakawa–Kanekotype,anddiscusstheirpropertiesandthe
relationwithpoly-Eulerpolynomials.Weestablishsomeresultstolaythefoun-
dationofpoly-Eulerpolynomialsandtheirassociatedzetafunctions.Therest
ofthispaperwillbeorganizedasfollows:weintroducepoly-Eulerpolynomials
andnumbersinSection2andpresentbasicresults.InSection3,weprovesome
theoremsstatedinSection2.InSection4,wedefinezetafunctionsofArakawa–
Kanekotype,whichareassociatedtopoly-Eulerpolynomials.InSection5,using
Dirichletcharacters,wegeneralizepoly-Eulerpolynomials,Arakawa–Kanekotype
zetafunctions,andrelatedresults.InSection6,weprovesometheoremsstated
inSection5.InSection7,wemakearemark.
2.Poly-Eulerpolynomials
2.1.Polylogarithms
Foranintegerk,letLik(x)betheformalpowerseriesgivenby
Lik(x)=
m=1
Σ
∞
mk
xm
.
(2.1)
Ifkisanegativeinteger,forinstancek=−r,thenitconvergesfor|x|<1and
equals
Lilr(x)=
Σ
(1−x)r+1
j=o(r
r
j>xrlj
j
(2.2)
wherethe(r
j>aretheEuleriannumbers.TheEuleriannumber(r
j>isthenumber
ofpermutationsof{1j...jr}withjpermutationascents.Onehas
<r
j>=
Σ
j+1
l=o
(−1)l(r+1
l)(j−l+1)r.
(2.3)
See[9]forsomepropertiesofEuleriannumbers.WegiveLilr(x)forsomer:
Lio(x)=
1−x
x
j
Lil2(x)=
(1−x)3
x2+x
j
Lil4(x)=
x4+11x3+11x2+x
(1−x)5
j
Lil1(x)=
(1−x)2
x
j
Lil3(x)=
x3+4x2+x
(1−x)4
j
Lil5(x)=
x5+26x4+66x3+26x2+x
(1−x)6
.
From(2.2)weimmediatelydeducethat,whenkisanegativeinteger,Lik(x)
isarationalfunctionwhosedenominatoris(1−x)'k'+1.Summingup,foran
