q-Euler Numbers and Polynomials Associated with Basic Zeta Functions

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

q-EULERNUMBERSANDPOLYNOMIALSASSOCIATEDWITHBASICZETAFUNCTIONS

arXiv:0807.2682v1 [math.NT] 17 Jul 2008

TaekyunKim

DivisionofGeneralEducation-Mathematics,KwangwoonUniversity,Seoul139-701,S.Korea

e-mail:tkkim@kw.ac.kr

Abstract.

Weconsidertheq-analogueofEulerzetafunctionwhichisde nedby

ζq,E(s)=[2]q

∞X( 1)nqnsn=1

Whenonetalksof

q-extension,qisvariouslyconsideredasanindeterminate,acomplexnumberq∈Corap-adicnumberq∈Cp.Ifq∈C,thenwenormallyassume|q|<1,andwhenq∈Cp,thenwenormallyassume|q 1|p<1.Weusethenotation:

[x]q=[x:q]=

1 qx

1+q

.

p.

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

2q-ANALOGUEOFRIEMANNZETAFUNCTION

Notethatlimq→1[x]q=xforx∈Zpinpresentedp-adiccase.

LetUD(Zp)bedenotedbythesetofuniformlydi erentiablefunctionsonZp.Forf∈UD(Zp),letusstartwiththeexpression

1

[pN] q

0≤j<pN

f(j)( q)j,(see[5,6,16]).

Forda xedpositiveintegerwith(p,d)=1,let

N

X=Xd=←lim Z/dpZ,X1=Zp,

N

X=

0<a<dp

(a,p)=1

a+dpZp,

a+dpNZp={x∈X|x≡a(moddpN)},

wherea∈Zliesin0≤a<dpN,(see[1-30]).

LetNbethesetofpositiveintegers.Form,k∈N,theq-Eulerpolynomials( m,k)Em(x,q)ofhigherorderinthevariablesxinCpbymakinguseofthep-adicq-integral,cf.[5,6],arede nedby

( m,k)

(1)Em,q(x)=

Zp

···

Zp

Zp

[x+x1+x2+···+xk]mq

·q

ktimes

x1(m+1) x2(m+2) ··· xk(m+k)

dµ q(x1)dµ q(x2)···dµ q(xk).

Now,wede netheq-Eulernumbersofhigherorderasfollows:

( m,k)( m,k)Em,q=Em,q(0).

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

3

From(1),wecanderive

( m,k)Em,q

(2)

=lim

1

m m

( 1)iii=0

N→∞

1

(1 q)m

et+1

k

=

n=0

∞

(k)tEn

n

(1 q)m

m

m

( 1)jqjx

jj=0

1

[n]sq

,q∈Rwith0<q<1ands∈C.

Thenumeratorensurestheconvergence.In(4),wecanconsiderthefollowingproblem:

“Arethereq-Eulernumberswhichcanbeviewedasinterpolatingofζq,E(s)atnegativeintegers,inthesamewaythatRiemannzetafunctioninterpolatesBernoullinumbersatnegativeintegers”?

Inthispaper,wegivethevalueζq,E( m),form∈N,whichisaansweroftheaboveproblemandconstructanewcomplexq-analogueofHurwitz’stypeEulerzetafunctionandq-L-seriesrelatedtoq-Eulernumbers.Also,wewilltreatsomeinterestingidentitiesofq-Eulernumbers.

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

),

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

5

By(3),(8),weeasilysee:

∞

Fq(t,x)=[2]q

(9)

k=0

1

1+qj k.

k

t

k!

Di erentiatingbothsideswithrespecttotin(5),(6)andcomparingcoe cients,

weobtainthefollowing:

Theorem1.Form≥0,wehave(10)

( m,1)Em,q(x)=[2]q

∞

n

q nm[n+x]mq( 1).

n=0

Corollary2.Letm∈N.Thenthereexists(11)

( m,1)

Em,q=[2]q

∞

n

q nm[n]mq( 1),andE0,q

(0,1)

=

[2]q

n=1

[2]qd[2]qd

[d]mq[d]mq

d 1 i=0d 1 i=0

q mi( 1)iχ(i)

q dx(m+1)[

( m,1)

i

Zp

χ(i)( 1)iq miEm,qd(

i

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

6q-ANALOGUEOFRIEMANNZETAFUNCTION

3.q-analogsofzetafunctions

Inthissection,weassumeq∈Rwith0<q<1.Nowweconsidertheq-extensionoftheEulerzetafunctionasfollows:

∞ns

nq( 1)ζq,E(s)=[2]q

n=1

[n+x]sq

.

Notethatζq,E(s,x)isananalyticcontinuationinwholecomplexs-plane.

By(14)andTheorem1,wehavethefollowingtheorem.Theorem4.Foranypositiveintegerk,wehave(15)

ζq,E( k,x)=Ek,q

( k,1)

(x,q).

Ford∈Nwithd≡1(mod2),letχbeDirichletcharacterwithconductord.By(13),thegeneralizedq-Eulernumbersattachedtoχcanbede nedas(16)

( m,1)Em,χ,q=

[2]q

d

).

Fors∈C,wede ne(17)

Lq,E(s,χ)=[2]q

∞ χ(n)( 1)nqsn

n=1

[2]qd

s

[d] q

a=1

d

χ(a)( 1)aqsaζqd,E(s,

a

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

7

Theorem5.Letk∈N.Thenthereexists

Lq,E( k,χ)=Ek,χ,q.

LetaandFbeintegerswith0<a<F.Fors∈C,weconsiderthefunctionsHq(s,a,F)asfollows:

Hq,E(s,a,F)=[2]q

qms( 1)m

[2]qF

saa[F] q( 1)qζqF(s,

( k,1)

a

m≡a(F),m>0

[2]qF

( n,1)

[F]nEqn,qF(

a

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

8q-ANALOGUEOFRIEMANNZETAFUNCTION

8.T.Kim,q-BernoullinumbersandpolynomialsassociatedwithGaussianbinomialcoe cients,RussianJ.Math.Phys.15(2008),51-57.9.T.Kim,q-extensionoftheEulerformulaandtrigonometricfunctions,RussianJ.Math.Phys.14(2007),275-278.10.T.Kim,J.y.Choi,J.Y.Sug,Extendedq-Eulernumbersandpolynomialsassociatedwithfermionic

p-adicq-integralonZp,RussianJ.Math.Phys.14(2007),160-163.11.T.Kim,q-generalizedEulernumbersandpolynomials,RussianJ.Math.Phys.13(2006),293-298.12.T.Kim,Multiplep-adicL-function,RussianJ.Math.Phys.13(2006),151-157.

13.T.Kim,Powerseriesandasymptoticseriesassociatedwiththeq-analogofthetwo-variablep-adic

L-function,RussianJ.Math.Phys.12(2005),186-196.14.T.Kim,Analyticcontinuationofmultipleq-zetafunctionsandtheirvaluesatnegativeintegers,

RussianJ.Math.Phys.11(2004),71-76.15.T.Kim,OnEuler-Barnesmultiplezetafunctions,RussianJ.Math.Phys.10(2003),261-267.16.T.Kim,Themodi edq-Eulernumbersandpolynomials,Adv.Stud.Contemp.Math.16(2008),

161-170.17.T.Kim,Anoteonp-adicq-integralonZpassociatedwithq-Eulernumbers,Adv.Stud.Contemp.

Math.15(2007),133-137.18.T.Kim,Anoteonp-adicinvariantintegralintheringsofp-adicintegers,Adv.Stud.Contemp.

Math.13(2006),95-99.19.H.Ozden,Y.Simsek,S.-H.Rim,I.N.Cangul,Oninterpolationfunctionsofthetwistedgeneralized

Frobenius-Eulernumbers,Adv.Stud.Contemp.Math.15(2007),187-194.20.H.Ozden,Y.Simsek,I.N.Cangul,Multivariateinterpolationfunctionsofhigher-orderq-Euler

numbersandtheirapplications,AbstractandAppliedAnalysis2008(2008),Art.ID390857,16pages.21.H.Ozden,Y.Simsek,I.N.Cangul,Eulerpolynomialsassociatedwithp-adicq-Eulermeasure,

GeneralMathematics15(2007),24-37.22.Y.Simsek,GeneratingfunctionsofthetwistedBernoullinumbersandpolynomialsassociatedwith

theirinterpolationfunctions,Adv.Stud.Contemp.Math.16(2008),251-278.23.Y.Simsek,Y.Osman,V.Kurt,OninterpolationfunctionsofthetwistedgeneralizedFrobenius-Eulernumbers,Adv.Stud.Contemp.Math.15(2007),187-194.24.Y.Simsek,HardycharactersumsrelatedtoEisensteinseriesandthetafunctions,Adv.Stud.

Contemp.Math.12(2006),39-53.

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

9

25.Y.Simsek,RemarksonreciprocitylawsoftheDedekindandHardysums,Adv.Stud.Contemp.

Math.12(2006),237-246.26.Y.Simsek,TheoremsontwistedL-functionandtwistedBernoullinumbers,Adv.Stud.Contemp.

Math.11(2005),205-218.27.Y.Simsek,D.Kim,S.-H.Rim,Onthetwo-variableDirichletq-L-series,Adv.Stud.Contemp.

Math.10(2005),131-142.28.Y.Simsek,A.Mehmet,RemarksonDedekindetafunction,thetafunctionsandEisensteinseries

undertheHeckeoperators,Adv.Stud.Contemp.Math.10(2005),15-24.29.Y.Simsek,Y.Sheldon,TransformationoffourTitchmarsh-typein niteintegralsandgeneralized

DedekindsumsassociatedwithLambertseries,Adv.Stud.Contemp.Math.9(2004),195-202.30.Y.Simsek,Onp-adictwistedq-L-functionsrelatedtogeneralizedtwistedBernoullinumbers,

RussianJ.Math.Phys.13(2006),340-348.

本文来源：https://www.bwwdw.com/article/f2r1.html