somemockthetafunctions
A.K.Agarwal∗
CentreforAdvancedStudyinMathematics
PanjabUniversityChandigarh-160014,IndiaE-mail:aka@pu.ac.in
Submitted:Mar13,2004;Accepted:Sep14,2004;Published:Sep20,2004MRSubjectClassifications(2000):Primary05A15,Secondary05A17,11P81.
Abstract
Usingn-colorpartitionsweprovidenewnumbertheoreticinterpretationsoffourmockthetafunctionsofS.Ramanujan.
1Introduction
InhislastlettertoG.H.Hardy,S.Ramanujanlisted17functionswhichhecalledmockthetafunctions.Heseparatedthese17functionsintothreeclasses.Firstcontaining4functionsoforder3,secondcontaining10functionsoforder5andthethirdcontaing3functionsoforder7.Watson[8]foundthreemorefunctionsoforder3andtwomoreoforder5appearinthelostnotebook[7].Mockthetafunctionsoforder6and8havealsobeenstudiedin[3]and[4],respectively.Forthedefinitionsofmockthetafunctionsandtheirorderthereaderisreferredto[6].Theobjectofthispaperistoprovidenewnumbertheoreticinterpretationsofthefollowingmockthetafunctions:
Ψ(q)=
∞m=1
qm
2
(q;q2)m
Φ0(q)=
∞m=0
2
,(1.2)(1.3)
qm(−q;q2)m,
and
Φ1(q)=
where
∞m=0
q(m+1)(−q;q2)m,
2
(1.4)
(a;q)n=
∞(1−aqi)i=0
Theorem3.Forν≥0,letA3(ν)denotethenumberofn-colorpartitionsofνsuchthatonlythefirstcopyoftheoddpartsandthesecondcopyoftheevenpartsareused,thatis,thepartsareofthetype(2k−1)1or(2k)2,theminimumpartis11or22,andtheweighteddifferenceofanytwoconsecutivepartsis0.Then,
∞ν=0
A3(ν)qν=Φ0(q).
(1.7)
Theorem4.Forν≥1,letA4(ν)denotethenumberofn-colorpartitionsofνsuchthat
onlythefirstcopyoftheoddpartsandthesecondcopyoftheevenpartsareused,theminimumpartis11,andtheweighteddifferenceofanytwoconsecutivepartsis0.Then,
∞ν=1
A4(ν)=Φ1(q).
(1.8)
Remark.Weremarkthatthereare160n-colorpartitionsof8butonlyonepartition
viz.,62+22isrelevantforTheorem3andnoneisrelevantforTheorem4.Outof859n-colorpartitionsof11,noneisrelevantforTheorems3-4.Among18334n-colorpar-titionsof17onlytwoviz.,91+62+22and82+51+31+11satisfytheconditionsofTheorem3,whereasthelonepartition82+51+31+11satisfiestheconditionsofTheoem4.Followingthemethodof[1],wegiveinournextsectionthedetailproofofTheorem1andtheshortestpossibleproofsfortheremainingtheorems.InthesequelAi(m,ν),(1≤i≤4),willdenotethenumberofpartitionsofνenumeratedbyAi(ν)intomparts,andweshallwrite∞∞
fi(z,q)=
ν=0m=0
Ai(m,ν)zmqν.
(1.9)
Inourlastsectionweillustratehowournewresultscanbeusedtoyieldnewcombinatorialidentities.
2Proofs
ProofofTheorem1.WesplitthepartitionsenumeratedbyA1(m,ν)intotwoclasses:(1)thosethatcontain11asapart,andthosethatcontainkk,(k>1)asapart.Followingthemethodof[1]itcanbeeasilyprovedthatthepartitionsinClass(1)areenumeratedbyA1(m−1,ν−2m+1)andinClass(2)byA1(m,ν−2m+1),andso
A1(m,ν)=A1(m−1,ν−2m+1)+A1(m,ν−2m+1).
From(1.9),wehave
f1(z,q)=
∞∞ν=0m=0
(2.1)
A1(m,ν)zmqν.
(2.2)
SubstitutingforA1(m,ν)from(2.1)in(2.2)andthensimplifyingweget
f1(z,q)=zqf1(zq2,q)+q−1f1(zq2,q).
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(2.3)
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Settingf1(z,q)=(2.3),weseethat
∞n=0
αn(q)zn,andthencomparingthecofficientsofznoneachsideof
αn(q)=
q2n−1
(q;q2)n
Therefore
f1(z,q)=
2
.(2.5)
∞qnznn=0
(q;q2)n
=Ψ(q).
ThiscompletestheproofofTheorem1.ProofofTheorem2.
TheproofissimilartothatofTheorem1,henceweomitthedetailsandgiveonly
theq-functionalequationusedinthiscase.
f2(z,q)=zq2f2(zq4,q)+q−1f2(zq,q).
ProofofTheorem3.
WesplitthepartitionsenumeratedbyA3(m,ν)intotwoclasses:(1)thosethatcontain11asapart,and(2)thosethatcontain22asapart.ByusingtheusualtechniqueweseethatthepartitionsinClass(1)areenumeratedbyA3(m−1,ν−2m+1)andinClass(2)byA3(m−1,ν−4m+2).Thisleadstotheidentity
A3(m,ν)=A3(m−1,ν−2m+1)+A3(m−1,ν−4m+2).
Using(2.8)onecaneasilyobtainthefollowingq-functionalequation
f3(z,q)=zqf3(zq2,q)+zq2f3(zq4,q).
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(2.7)
(2.8)
(2.9)
4
Settingf3(z,q)=
∞n=0
βn(q)zn,andnotingthatf3(0,q)=1,wecaneasilycheckby
coefficientcomparisonin(2.9)that
βn(q)=qn(−q;q2)n.
Therefore,
f3(z,q)=
Now
∞ν=0
∞n=0
22
(2.10)
qn(−q;q2)nzn.
A3(m,ν))qν
(2.11)
A3(ν)q
ν
=
∞∞
(
ν=0m=0
=f3(1,q)=
∞n=0
qn(−q;q2)n
2
=Φ0(q).
ThisprovesTheorem3.ProofofTheorem4.
ThepartitionsenumeratedbyA4(m,ν)arepreciselythosepartitionswhichbelongto
Class1ofthepreviouscase.Therefore,
A4(z,ν)=A3(m−1,ν−2m+1).
(2.12)
UsingEquations(2.8)and(2.12),onecaneasilyobtainthefollowingq-functionalequation:
f4(z,q)=f3(z,q)−zq2f3(zq4,q).
Settingf4(z,q)=
∞n=0
(2.13)
γn(q)zn,andthencomparingthecoefficientsofznoneachsideof
γn(q)=βn(q)−βn−1(q)q4n−2
=qn(−q;q2)n−1.
2
(2.13),weseethat
Thisimpliesthat
f4(z,q)=
∞n=1
qn(−q;q2)n−1zn.
2
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Now
∞ν=0
A4(ν)q
ν
=
∞∞
(
ν=0m=0
A4(m,ν))qν
=f4(1,q)==
∞n=1
∞n=0
qn(−q;q2)n−1q(n+1)(−q;q2)n
2
2
=Φ1(q).
ThiscompletestheproofofTheorem4.
3Newcombinatorialidentities
OurTheorems1-4canbecombinedwiththeknownnumbertheoreticinterpretationsof(1.1)-(1.4)toyieldnewcombinatorialidentities.Forexample,Theorem1inviewoftheknownpartitiontheoreticinterpretationofΨ(q)givenaboveinSection1givesthefollowingresult:
Theorem5.Forν≥1,thenumberofn-colorpartitionsofνsuchthatevenpartsappearwithevensubscriptsandoddwithodd,forsomek,kkisapart,andtheweighteddifferenceofanytwoconsecutivepartsis0equalsthenumberofordinarypartitionsofνintooddpartswithoutgaps.
References
1.A.K.Agarwal,Rogers-Ramanujanidentitiesforn-colorpartitions,J.NumberThe-ory28(1988),299–305.
2.A.K.AgarwalandG.E.Andrews,Rogers-Ramanujanidentitiesforpartitionswith“NcopiesofN”,J.Combin.TheorySer.A45,(1987),40–49.
3.G.E.AndrewsandD.Hickerson,Ramanujan’s”Lost”NotebookVII:Thesixthor-dermockthetafunctions,Adv.Math.,89(1991),60–105.
4.B.GordonandR.J.McIntosh,Someeightordermockthetafunctions,J.LondonMath.Soc.(2)62(2000),321–335.
5.N.J.Fine,BasicHypergeometricSeriesandApplications,MathematicalSurveysandMonographs,No.27,AMS,(1988)
6.G.H.HardyandE.M.Wright,AnIntroductiontotheTheoryofNumbers,FifthEdition,OxfordUniversityPress(1978).
7.S.Ramanujan,TheLostNotebookandotherUnpublishedPapers,NarosaPublish-ingHouse,NewDelhi,1988
8.G.N.Watson,Thefinalproblem:anaccountofthemockthetafunctions,J.LondonMath.Soc.,11(1936),55–80.
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