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1、P1:ABLP1:ABLTHERAMANUJANJOURNALKl616-01-SonAugust11,199815:54THERAMANUJANJOURNAL2,303Ð316(1998)°c1998KluwerAcademicPublishers.ManufacturedinTheNetherlands.CubicIdentitiesofThetaFunctionsSEUNGH.SONson@math.uiuc.eduDepartmentofMathematics,UniversityofIllinois,Urbana,Illinois61801Receiv
2、edJuly2,1996;AcceptedApril28,1997Abstract.ManyremarkablecubictheoremsinvolvingthetafunctionscanbefoundinRamanujanÕsLostNote-book.Usingadditionformulas,theJacobitripleproductidentityandthequintupleproductidentity,weestablishseveraltheoremstoproveRamanujanÕscubicidentities.Keywords:the
3、tafunction,Jacobitripleproductidentity,EulerÕspentagonalnumbertheorem,residueclasses1991MathematicsSubjectClassiÞcation:PrimaryÑ33D101.IntroductionRamanujanÕssymmetricthetafunctionf.a;b/isdeÞnedbyX1f.a;b/:Dak.kC1/=2bk.k¡1/=2;jabj<1;kD¡1whichhasthesamegeneralityastheclassicalthetafunc
4、tionsfµ.z;q/g4in[7,pp.463,iiD1464].Forbrevity,weneedtodeÞnef.¡q/:Df.¡q;¡q2/andÃ.q/:Df.q;q3/.Onpages48and54inhisLostNotebook[4],Ramanujanrecordedmanycubicidenti-tiesinvolvingcertainthetafunctions.Berndt[3,pp.142Ð146],[2]provedseveralelegantidentities,includingf3.a3b6;a6b3/Ca3f3.b3;a9b
5、6/Cb3f3.a3;a6b9/f.a3;b3/f3.¡ab/f.a3;b3/f3.¡a9b9/DC3ab(1.1)f.¡a3b3/f.¡a3b3/µ993¶1=3f.¡x/f.¡x/Df.a3;b3/C27x(1.2)f3.¡x3/f3.¡x/µ¶Ã3.x/Ã3.x3/Df.a3;b3/C3x(1.3)Ã.x3/Ã.x/andf.a3;b3/ff.a;b/¡f.a3b6;a6b3/g3Df3.ab2;a2b/¡f3.a3b6;a6b3/;f.a3b6;a6b3/wherex:Da3b3.Theseevincebeautifulsymmetry.P1:ABLP1
6、:ABLTHERAMANUJANJOURNALKl616-01-SonAugust11,199815:54304SONInthispaperweestablishseveraltheoremsinordertoproveRamanujanÕscubicidentitiesfoundinhisLostNotebook;ourmethodsalsoyieldsomecubicmodularequations.Toprovethetheorems,weemployageneralthetafunctionproductformula,amethodusingkerne
7、lsfromsummandsofthetafunctionsandcubicidentitiessuchas(1.1)aswellasseveralfamoustheorems.2.PreliminaryresultsForjqj<1;setnY¡1Y1.aIq/:D.1¡aqk/and.aIq/:D.1¡aqk/:n1kD0kD0WeshallfrequentlyusetheJacobitripleproductidentity(2.1).Theorem2.1.Forjabj<1;wehavef.a;b/D.¡aIab/1.¡bIab/1.abIab/1:(2
8、.1)Foraproof
