Ramanujan's Theta Functions

696 Pages·2017·6.408 MB·English

by  Shaun Cooper (auth.)

Shaun Cooper
Ramanujan’s Theta
Functions
Ramanujan’s Theta Functions
Shaun Cooper
Ramanujan’s Theta
Functions
123
ShaunCooper
InstituteofNaturalandMathematical
Sciences
MasseyUniversity
Auckland,NewZealand
ISBN978-3-319-56171-4 ISBN978-3-319-56172-1 (eBook)
DOI10.1007/978-3-319-56172-1
LibraryofCongressControlNumber:2017937318
Mathematics Subject Classification: 11-02 (primary); 05A30, 11A55, 11F11, 11F27, 11Y55, 33-02
(secondary)
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ToHilary
Preface
Athetafunctionisaseriesoftheform
1
X qn2xn; where jqj<1 and 0<jxj<1;
nD(cid:2)1
orageneralizationorspecializationofsuchaseries.Thetafunctionswerestudied
extensivelybyRamanujanandformthecentralthemeofthisbook.
Quotients of theta functions can be used to construct elliptic functions and
modular functions. One of the incredible results discovered by Jacobi is that the
quotientofthetafunctionsdefinedby
0 1 14
X q.nC12/2
B C
xDBBnD(cid:2)1 CC for 0<q<1 (1)
B 1 C
@ X qn2 A
nD(cid:2)1
hasaninversefunctionthatisgivenbytheformula
(cid:2) F.1(cid:2)x/(cid:3) X1 2n!2 x !n
qDexp (cid:2)(cid:2) ; where F.x/D (2)
F.x/ n 16
nD0
!
2n
and isthebinomialcoefficient.Moreover,
n
1 !2 !n 1 !2
X 2n x D X qn2 : (3)
n 16
nD0 nD(cid:2)1
vii
viii Preface
Jacobi’s results are fundamental to the theory, computation, and applications of
Jacobianellipticfunctionsandellipticintegrals.
RamanujandiscoveredthreeanaloguesofJacobi’sresult.Atthetime,theybarely
received attention. But since the 1990s they have become the subject of intense
interest.
The goal of this book is to provide a systematic development of the results of
JacobiandRamanujanandtoextendtheresultstoageneraltheory.Thetreatment
is comprehensive and provides a detailed study of theta functions and modular
forms for levels up to 12. Jacobi’s results in (1)–(3) above arise in the theory for
level 4, while Ramanujan’s alternative results occur in the corresponding theories
for levels 1, 2, and 3. The organization of the material lends itself to serving as
a useful reference. Additionally, the book contains a rich source of examples for
theoreticians in number theory and modular forms. The topics, especially those in
thesecondhalfofthebook,havebeenthesubjectofmuchrecentresearch;manyof
thesealsoappearingforthefirsttimeinbookform.Furtherresultsaresummarized
inthenumerousexercisesattheendofeachchapter.
Thisbookhasbeenwrittenforadvancedundergraduates,graduatestudents,and
researchers. It is user friendly and in principle can be used to learn the subject
from scratch. The prerequisites are fairly straightforward and rely mainly on an
understandingofundergraduatecomplexanalysis(especiallyintermsofthezeros
andpolesofmeromorphicfunctions,andconvergenceofinfiniteseriesandinfinite
products)andthebasictheoryoflineardifferentialequations.Thereaderisexpected
to come up to speed quickly with manipulations of series and products; that is
inevitableinanyseriousstudyofRamanujan’smathematics.
Chapter0containsanintroductiontothemainidentitiesinq-analysis:Jacobi’s
tripleproductidentity,thequintupleproductidentity,Ramanujan’ssummationfor-
mula,andtheq-binomialtheorem.Wealsomeetq-analoguesofthesineandcosine
functions,whicharedenotedbys.(cid:3)/andc.(cid:3)/,respectively.Thefunctions.(cid:3)/has
zerosthataresimpleandlieonalattice,andbecauseofthisfundamentalstructure,
itplaysanimportantrolethroughoutthebook.
Chapter 1 develops the theory of elliptic functions to deduce identities that can
beusedtoconvertinfiniteproductstoseries(andviceversa).Specificexamplesthat
leadtothenumberofrepresentationsofanintegerbythequadraticformsx2C7y2
andx2Cy2C3z2C3w2areworkedthroughindetail.PropertiesoftheWeierstrass,
Jacobian, and Dixon elliptic functions are developed from q-infinite products, the
latterbeingthefirst(totheauthor’sknowledge)inabook.
Chapter2containsanintroductiontomodulartransformationswithanemphasis
onexamples.Mostbookspresentonlythesimplestexamplesandexpendaconsider-
ableamountoflabortohandleconditionallyconvergentseries.WetreatEisenstein
seriesthatarisefromeitherthemodulargroup(cid:4) ortheHeckecongruencesubgroup
(cid:4) .p/caseinasimplewaybystartingwithabsolutelyconvergentinfiniteproducts.
0
Theanalysisalsoworksformodularformsofweightsoneandtwoandavoidsthe
difficultiesofconditionalconvergencethatariseinthemorestandardapproaches.
Preface ix
Chapter 3 provides a comprehensive and systematic treatment of the main
propertiesofRamanujan’sthetafunctions
1 1
'.q/D X qj2 and .q/DXqj.jC1/=2
jD(cid:2)1 jD0
andtheBorweins’cubicthetafunctionsa.q/,b.q/,andc.q/.Theapproachempha-
sizes the interconnections between the two sets of functions and pays attention to
how properties for one set of functions carry over to the other set of functions.
The identities are classified by weight, and some have not been published before.
Thechapterendswithananalysisofrepresentationsofintegersbyquadraticforms
and includes results for sums of 2, 4, 6, 8, 10, and 12 squares (and also triangular
numbers)aswellasadiscussionofRamanujan’sgeneralresultforsumsofanyeven
numberofsquaresandanyevennumberoftriangularnumbers.Cubicanaloguesare
also analyzed. Exercises include representations by several other quadratic forms,
includingx2Cmy2 wherem D 2,3,4,5,6,and15.Thecasesm D 7andm D 8
aredetailedinChapters1and8,respectively.
In Chapter 4, the theory of theta functions is used to prove Jacobi’s inversion
theorem given by (1)–(3). The same procedure is used to prove the main results
for Ramanujan’s alternative theories. The most significant result is Theorem 4.30,
in which all four theories occur in a unified way and where F hypergeometric
3 2
functionsareparameterizedbythetafunctions.Thearithmetic-geometricmeanand
otheriterationsareanalyzedasaconsequence,andhypergeometrictransformation
formulasareobtained.
In Chapter 5 we start with the Rogers–Ramanujan continued fraction and
establish its expansion as an infinite product. We prove the Rogers–Ramanujan
identitiesandRamanujan’spartitioncongruencemodulo5.Fromthere,itisshown
that the Rogers–Ramanujan continued fraction is central to a level 5 theory that
is analogous to Jacobi’s results in (1)–(3). This gives rise to Apéry-type numbers
andZagier’ssporadicsequences,andalsototwostunningandrecentanaloguesof
Clausen’sidentityforthesquaresofcertainpowerseries.
Chapter 6 centers around Ramanujan’s cubic continued fraction. The results in
this chapter are the level 6 analogues of the results in Chapter 5. An interesting
aspect is that there are three analogues in the level 6 theory of every function for
level5.
The corresponding theories for levels 7–12 are developed in Chapters 7–12,
respectively. Each theory has its own idiosyncrasies and individual features. The
results in Chapter 11 reach the limits of our approach, which up to this point has
beenbythetafunctions.Weobtainaglimpseintotheimportantroleofthetheoryof
modularformsinstudyinghigherlevels.Thetheoryforlevel23canbedeveloped
withnoextraeffortandisalsoincluded.
x Preface
In Chapter 13, the level 10 and level 12 theories are used to develop and sys-
tematizeanenormousnumberofhypergeometricmodulartransformationformulas.
Anexampleofoneofthebeautifulspecialcasesthatarisefromtheidentitiesinthis
chapteris
1 (cid:2)1; 1; 1 (cid:2) 1Ct (cid:3)5(cid:3)
F 2 2 2I(cid:2)64t
.1(cid:2)4t/5=2 3 2 1; 1 1(cid:2)4t
1 (cid:2)1; 1; 1 (cid:2) 1Ct (cid:3)(cid:3)
D F 2 2 2I(cid:2)64t5
.1(cid:2)4t/1=2 3 2 1; 1 1(cid:2)4t
8 9
1 X1 <Xn n!4= t.1Ct/.1(cid:2)4t/!n
D :
.1C4t2/3=2 : j ; .1C4t2/2
nD0 jD0
Finally, in Chapter 14 we show how everything in the book can be brought
together to explain a class of series for 1=(cid:2), which are due to Ramanujan. This
culminatesinTables14.1and14.2,whichprovideaconcisesummaryofthemain
results.
My teacher Richard Askey opened the door to Ramanujan’s mathematics to
me and I have been captivated by the subject ever since. My career has been
enriched through interactions with coauthors, and I especially acknowledge Heng
Huat Chan, Michael Hirschhorn, and my recent student Dongxi Ye. Chapters 0–4
havebeenusedasthebasisofaone-semestergraduatecourseatMasseyUniversity
since2001.Iamgratefultoallofthestudentswhohaveparticipatedinthecourse
and especially to those who have gone on to write theses, including Uros Abaz,
AntesarAldawoud,JinqiGe,HeungYeungLam,LynetteO’Brien,andDongxiYe.
Themanuscriptwaspreparedoveranumberofyearsandwascompletedduringlong
leaveattheUniversityofNewcastleinAustralia.MichaelCoons,MumtazHussain,
Wadim Zudilin, and the late Jonathan Borwein provided excellent hospitality and
a wonderful working environment at Newcastle. Invigorating discussions with
Zudilin helped shape the last two chapters. Hilary Boyd, Jesús Guillera, Michael
Hirschhorn, Winston Sweatman, and Pee Choon Toh supplied feedback on parts
of the manuscript. Elizabeth Loew at Springer provided enthusiastic and expert
guidance to bring the manuscript to publication. To all of the people mentioned
inthisparagraph,Ioffermyheartfeltthanks.
Auckland,NewZealand ShaunCooper
February2017
Contents
0 SumtoProductIdentities.................................................. 1
0.1 Introduction ......................................................... 1
0.2 Euler’sidentityforinfiniteproducts ............................... 1
0.3 Jacobi’stripleproductidentity ..................................... 4
0.4 Ramanujan’sthetafunctionf.a;b/................................. 8
0.5 Thequintupleproductidentity ..................................... 10
0.6 Examples............................................................ 15
0.7 Partitions ............................................................ 18
0.8 q-analoguesofsineandcosine..................................... 21
0.9 Ramanujan’sdifferentialequations ................................ 25
0.10 Ramanujan’ssummationformula.................................. 32
0.11 SpecialcasesofRamanujan’ssummationformula................ 37
0.12 Notes................................................................. 42
0.13 Exercises ............................................................ 44
1 EllipticFunctions........................................................... 59
1.1 IntroductionandtheWeierstrassellipticfunction................. 59
1.2 Generalpropertiesofellipticfunctions............................ 63
1.3 Ellipticfunctionidentities.......................................... 65
1.4 ExplicitformsofHalphen’sidentity............................... 69
1.5 ExplicitformsofWeierstrass’identity............................. 74
1.6 ExplicitformsofTheorems1.8,1.10,and1.11................... 76
1.7 Examples............................................................ 78
1.8 Jacobianellipticfunctions.......................................... 84
1.9 Differentialequationsandadditionformulas...................... 92
1.10 Dixon’sellipticfunctions........................................... 102
1.11 Dixon’sellipticfunctions:specialcase(cid:5) D1..................... 110
1.12 Dixon’sellipticfunctions:specialcase(cid:5) D0..................... 110
1.13 Notes................................................................. 113
1.14 Exercises ............................................................ 115
xi