Ramanujan's Series for 1/π: A Survey
Author(s): Nayandeep Deka Baruah, Bruce C. Berndt and Heng Huat Chan
Source: The American Mathematical Monthly, Vol. 116, No. 7 (Aug. - Sep., 2009), pp. 567-587
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/40391165
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Ramanujan'sSeriesfor1/tt:A Survey*
NayandeepDeka Baruah,BruceC. Berndt,and HengHuat Chan
In Memoryof V RamaswamyAiyer,
FounderoftheIndianMathematicalSocietyin 1907
When we pause to reflecton Ramanujan'slife,we see thattherewere certain
eventsthatseeminglywerenecessaryin orderthatRamanujanandhismathematOne of thesewas V. RamaswamyAiyer'sfounding
ics be broughtto posterity.
oftheIndianMathematicalSocietyon 4 April1907,forhad he notlaunchedthe
IndianMathematicalSociety,thenthenextnecessaryepisode,namely,Ramanuin 1910,would
jan's meetingwithRamaswamyAiyerathisofficeinTirtukkoilur
also havenottakenplace. Ramanujanhad carriedwithhimone ofhisnotebooks,
and RamaswamyAiyernotonlyrecognizedthecreativespiritthatproducedits
buthe also had thewisdomto contactothers,suchas R. Ramachandra
contents,
to othersforappreciationand
in
order
to bringRamanujan'smathematics
Rao,
that
The
mathematical
community has thrivedon Ramanujan's
large
support.
owes a hugedebtto V. RamaswamyAiyer.
discoveriesfornearlya century
1. THE BEGINNING. Toward the end of the firstpaper [57], [58, p. 36] that
Ramanujanpublishedin England,at thebeginningof Section 13, he writes,"I shall
concludethispaperby givinga few seriesfor1/7T."(In fact,Ramanujanconcluded
for
his papera couple of pages laterwithanothertopic:formulasand approximations
of an ellipse.) Aftersketchinghis ideas, whichwe examinein detail
the perimeter
for '/it. As is
in Sections3 and 9, Ramanujanrecordsthreeseriesrepresentations
set
customary,
(fl)o:= 1,
(a)n := a(a + 1) • • • (a + n - 1),
n > 1.
Let
(1)3
A-:=% n!3
n>0.
(1.1)
Theorem1.1. IfAnis defined
(1.1),then
fry
l = ¿(6n +
l)An^,
7
(1.2)
= £(42«+
5Mn¿,
(1.3)
/î=0
' 8n
^r1 •
(1-4)
thefounding
of
Studenttocommemorate
solicitedbytheEditorofMathematics
This paperwas originally
Studentis one ofthetwoofficial
theIndianMathematical
journals
year.Mathematics
Societyin itscentennial
Society,withtheotherbeingtheJournalof theIndianMathematical
publishedby theIndianMathematical
Society.The authorsthankthe Editorof MathematicsStudentforpermissionto reprintthe articlein this
Monthly withminorchangesfromtheoriginal.
2009]
August-September
ramanujan' s series for i/n
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567
The first
twoformulas,
(1.2) and (1.3), appearedin theWaltDisneyfilmHigh
SchoolMusical,starring
VanessaAnneHudgens,
whoplaysan exceptionally
bright
student
that
school
named
Gabriella
Montez.
Gabriella
high
pointsouttoherteacher
shehadincorrectly
written
theleft-hand
sideof(1.3) as S/n insteadof 16/non the
blackboard.
After
first
checks(possibly
thatGabriellais wrong,
herteacher
claiming
Collected
and
admits
that
Gabriella
is
correct.
Formula(1.2)
Ramanujan's
Papers!)
wascorrectly
recorded
ontheblackboard.
After
thethreeformulas
for'/n givenabove,atthebeginning
ofSection
offering
14 [57],[58,p. 37],Ramanujan
in
"There
are
theories
whichq
claims,
corresponding
is replaced
one
or
other
of
the
functions"
by
(- - • I* 1 - jci'
-7Tcsc(7T/r)2
d-5)
M ,
lpnrrV.Vx)
(F
wherer = 3, 4, or 6, and where2^1 denotesone of thehypergeometric
functions
>
which
are
defined
1,
by
pFp-i,p
xn
it
/
u
u
'
V^ (a')n'-(ap)n
a'-bi-pFp-i{ai
Vi;*):=
•(»,.,).*■w<1S (»,),-
ofq is explained
inSection3.) Ramanujan
14further
thenoffers
series
(Themeaning
for'/n. Of these,10 belongto thequartictheory,
i.e.,forr = 4; 2
representations
i.e.,forr = 3; and2 belongto thesextictheory,
i.e.,for
belongto thecubictheory,
r - 6. Ramanujan
neverreturned
to the"corresponding
in his published
theories"
these
papers,butsix pagesin his secondnotebook[59] are devotedto developing
withall of theresultson thesesix pagesbeingprovedin a paper[16] by
theories,
S. Bhargava,
andF. G. Garvan.Thattheclassicalhypergeometric
function
Berndt,
in
the
classical
of
functions
could
be
oneof
1;
jc)
theory
elliptic
replaced
by
2Fi(^, ''
thethree
functions
aboveandconcomitant
theories
is oneof
hypergeometric
developed
themanyincredibly
andusefulideasbequeathed
tous byRamanujan.
The
ingenious
ofthesetheories
is farfromeasyandis an activeareaofcontemporary
development
research.
All 17 seriesfor'/n werediscovered
in Indiabeforehe arrived
by Ramanujan
inEngland,
fortheycanbe foundinhisnotebooks
[59],whichwerewritten
priorto
forEngland.In particular,
(1.2),(1.3),and(1.4) canbe found
Ramanujan's
departure
onpage355inhissecondnotebook
andtheremaining
14 seriesarefoundinhisthird
notebook[59,p. 378]; see also [14,pp. 352-354].It is interesting
that(1.2), (1.3),
and(1.4) arealso locatedon a pagepublished
withRamanujan's
lostnotebook
[60,
15].
p. 370];see also [3,Chapter
2. THE MAIN ACTORS FOLLOWING IN THE FOOTSTEPS OF RAMANUthepublication
of[57],thefirst
JAN.Fourteen
mathematician
toaddress
yearsafter
formulas
was
Sarvadaman
Chowla
[37],[38],[39,pp.87-91,116-119],
Ramanujan's
whogavethefirst
fori/n andused
published
proofofa generalseriesrepresentation
it to derive(1.2) of Ramanujan's
seriesforl/n [57,Eq. (28)]. We briefly
discuss
Chowla'sideasinSection4.
until
Ramanujan'sserieswerethenforgotten
by themathematical
community
Jr.
used
one
of
November,
1985,whenR. William
series,
Gosper,
Ramanujan's
namely,
9801 _
^
nVS~h
568
(4m)!(1103+ 26390ft)
(*!)43964*
'
( }
© THE MATHEMATICALASSOCIATION OF AMERICA [Monthly 116
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tocalculate17,526,100
Therewas
digitsofn, whichatthattimewasa worldrecord.
his
one
with
calculation
had
not
been
a
(2.1)
problem
yet
proved.However,
only
calculation
of
of
n
of
the
with
the
world
record
digits
previous
comparison Gosper's
that(2.1) wasincorrect.
heldbyY. Kanadamadeitextremely
unlikely
andPeterBorwein[23] succeededinproving
In 1987,Jonathan
all 17 ofRamanuIn
of
series
for
a
series
'/n.
subsequent
papers[24],[25],[29],theyestablished
jan's
seriesfor1/jr,withoneoftheirseries[29] yielding
severalfurther
fifty
roughly
digthedigitsofn9 and
itsofn perterm.The Borweinswerealso keenon calculating
oftheir
workcanbe foundin [30],[28],and[26].
accounts
weredevising
their
DavidandGreAtaboutthesametimeas theBorweins
proofs,
for
in particular,
also
derived
series
and,
[40]
representations'/n
goryChudnovsky
usedtheirseries
I = ilfv
*
ht
IV
(6n)!
13591409+ 545140134W
("!)3(3")!
(640320)3"+3/2
l *}
tocalculatea worldrecord2,260,331,336
digitsofn. Theseries(2.2) yields14digits
canbe foundina
calculations
A popularaccountoftheChudnovskys'
ofn perterm.
New
Yorker
written
for
The
[56].
paper
S. H. Chan,A. Gee,
author
ofthepresent
Thethird
(Berndt,
paperandhiscoauthors
ina seriesofpapers[19],[31],[33],[34],
W.-C.Liaw,Z.-G.Liu,V.Tan,andH. Verrill)
without
inparticular,
theideasoftheBorweins,
usingClausen'sformula
[36]extended
formulas
in[31]and[36],andderived
general
hypergeometric-like for1/n.Wedevote
results.
someoftheir
todiscussing
Section8 ofoursurvey
thefirst
twoauthors
the
thirdauthor,
of
work
and
the
Stimulated
suggestions
by
in
and
to
returned
[57]
employed
Ramanujan's
development
[9], [7] systematically
for
hisideasinordernotonlytoprovemostofRamanujan's
original
representations
In
as
well.
another
identities
such
of
new
a
establish
but
also
to
paper
plethora
i/rc
and
[8],motivated
bytheworkofJesusGuillera[48]-[53],whobothexperimentally
authors
the
first
two
and
both
new
series
for
discovered
1/7T2,
l/n
many
rigorously
fori/n2.
ideasanddevisedseriesrepresentations
tofollowRamanujan's
continued
themainideasin Sections3, 6, 7, 8,
In thesurveywhichfollows,we delineate
theChudnovsky
theBorwein
and9, wheretheideasofRamanujan,
brothers,
brothers,
arediscussed.
andthepresent
Chanandhiscoauthors,
authors,
respectively,
3. RAMANUJAN'SIDEAS. To describe
ideas,we needseveraldefiRamanujan's
infact,weusethroughout
ofelliptic
theclassicaltheory
from
nitions
which,
functions,
kindis defined
ofthefirst
thepaper.Thecomplete
by
ellipticintegral
K := K{k) := f
Jo
dv
,
.
vl - k2sin (p
(3.1)
K' := K(kf)ywherek' :=
wherek, 0 < k < 1, denotesthemodulus.Furthermore,
ofthesecond
The
modulus.
the
-k2
is
Vl
ellipticintegral
complete
complementary
kindis defined
by
E := E(k) :=
f7T/2 /
sin2<pd<p.
Jo y/l-k2
2009] ramanujan's seriesfor '/n
August-September
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(3.2)
569
in thetheory
of elliptic
If q = àxpi-nK'/K), thenone of thecentral
theorems
functions
asserts
that[13,p. 101,Entry
6]
<p'q) = ^K(k) = 2FX(','' 1; k2),
(3.3)
where<p(q)in Ramanujan's
notation
denotesthe
(or û^(q) in theclassicalnotation)
classicalthetafunction
defined
by
00
via)= J29jly=-oo
(3-4)
Notethat,inthenotation
in (3.3) fol(1.5),q- q2 andx = k2.The secondequality
lowsfrom
the
in
series
and
termwise.
Cona
binomial
expanding integrand
integrating
=
it
is
to
£
of
and
so
we
k
also
valuable
as
a
function
write
k(q).
versely,
q,
regard
Let K, K ', L, andV denotecomplete
ofthefirst
kindassociated
ellipticintegrals
withthemodulik,k' I, and£',respectively.
forsomepositive
n9
integer
Supposethat,
i'
v
A modular
k andI thatis inducedby
equationofdegreen is an equationinvolving
An exampleof a modular
(3.5). Modularequationsarealwaysalgebraicequations.
equationofdegree7 maybe foundlaterin (9.18). Alternatively,
by(3.3), (3.5) can
be expressed
interms
ofhypergeometric
Weoftensaythati hasdegreen
functions.
overk. Derivations
ofmodular
reston (3.3). If we setK'/K =
equations
ultimately
valueof¿, whichis denoted
y/ñ,so thatq = e~n^, thenthecorresponding
bykn:=
modulus.
Themultiplier
m = m(q) is defined
k(e~n^"),is calledthesingular
by
'
m := m(q) := 2F1(i,l;l;fc2(<?))
y '
(3.6)
Wenoteherethat,
3, p. 98; Entry
25(vii),p. 40],m(q)
by(3.3),(3.6),and[13,Entry
andk2{q)canbe represented
by
(p2(q)
rn{q)= ^^~
and
~
V{q2)
k2(q) = 16q^-f,
wherecp(q)is defined
respectively,
by(3.4) and
oo
y=0
canalsobe written
as thetafunction
identities.
Thus,modular
equations
Ramanujan
beginsSection13of[57]witha specialcaseofClausen'sformula
[23,
5.6(b)],
p. 178,Proposition
AK2
°°
C-Ì3
=
= ^ G.è.è;l i;^')2) E
ir
7fï^2**'>2y
570
(3-7)
(E) THE MATHEMATICALASSOCIATION OF AMERICA [Monthly 1 16
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whichcan be foundas Entry13 of Chapter11 in his secondnotebook[59] [12, p. 58].
Exceptforeconomizingnotation,we nowquoteRamanujan."Hence we have
v 2/3 oo /U3
/,
where
(a; <7)oo:= (1 - a) (I - aq)(l - aq2) • • • .
(3.9)
bothsidesin (3.8) withrespecttok,we can easilyshew
differentiating
Logarithmically
that
1- 24¿ J^Lj = (1 - 2k2)¿(3; +
')^±{2kk')2L
'J '*
^
7=0
7= 1
(3.10)
But itfollowsfrom
o
oo
l--^-24V
- 4
•
/ K'2
= (-)
A(t)
(3.11)
whereA (it) is a certaintypeof algebraicnumber,that,whenq = e~n^", n being a
side of (3.10) can be expressedin theform
theleft-hand
rationalnumber,
B
A (2K'2
+-,
n
'n )
whereA and B are algebraicnumbersexpressibleby surds.Combining(3.7) and
(3.10) in such a way as to eliminatetheterm(2K/n)2, we are leftwitha seriesfor
He thengivesthethreeexamples(1.2)- (1.4).
1/7T."
Ramanujan'sideas will be describedin moredetailin Section9. However,in clossides of (3.10) and (3.11) is
ing thissection,we notethattheserieson theleft-hand
=
in
thelatterinstance,where
e'71^
with
series
Eisenstein
q
P(q2),
Ramanujan's
P(q) := 1-24¿-^-,
7= 1
*
'q' < 1.
(3.12)
formulafor
fromthe transformation
Ramanujan'sderivationof (3.11) arises firstly
forthe
formula
of
the
transformation
an
in
is
which
turn
easy consequence
P(q),
in
The
second
in
below.
Dedekindeta-function,
ingredient deriving(3.1 1)
given (9.1 1)
fornP{q2n) - P(q2) in termsof themodulik and I, wherei has degree
is an identity
n over k. Formula(3.8) followsfroma standardtheoremin ellipticfunctionsthat
Ramanujanalso recordedin his notebooks[59], [13, p. 124,Entry12(iii)].
4. SARVADAMAN CHOWLA. Chowla's ideas reside in the classical theoryof
ellipticfunctionsand are notunlikethosethattheBorweinsemployedseveralyears
later.We now brieflydescribeChowla's approach[37], [38], [39, pp. 87-91, 1loll 9]. Using classical formulasof Cayleyand Legendrerelatingthecompleteelliptic
2009]
August-September
ramanujan's series for i/n
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571
he specializesthemby
K and £, defined
integrals
by (3.1) and (3.2), respectively,
=
He
then
defines
K/K' y/n.
setting
S'^É/M*2**')2'
(41)
Tr:=f^r^2(2kkf)2J.
V "'
(4.2)
and
j=0
Chowlathenwrites
"Thenitis known
whenk < l/'/2,"
that,
-2K = l + r0
n
and
4K2
-= l + 50.
re2
(4.3)
in(4.3),as
Chowladoesnotgivehissourceforeither
butthesecondformula
formula,
formula
is
a special
notedabove,is a specialcaseofClausen'sformula
The
first
(3.7).
was
also
known
caseofKummer's
which
transformation
[23,pp. 179-180],
quadratic
toRamanujan.
tok,
Eachoftheformulas
of(4.3) is differentiated
twicewithrespect
=
if
then
without
concludes
that
Chowla
and,
K/Kr y/n,
givingdetails,
- =ai +biTo + ciTu
ä
- =dxTx+exT2,
71
- =a2S0-'-b2Sìì
TC
-
= 03 + b3So + C3S1 + ¿Z3S2,
"whereax, b', . . . arealgebraic
numbers."
He thensetsn = 3 andk = sin(7r/12)
in
eachofthefourformulas
abovetodeduce,inparticular,
thesecond
(1.2) from
identity
formula
above.
5. R. WILLIAM GOSPER, JR. As we indicated
intheIntroduction,
inNovember,
a lispmachine
at Symbolics
andRamanujan's
series(2.1) to
1985,Gosperemployed
calculate17,526,100
digitsof7r,whichat thattimewas a worldrecord.(Duringthe
1980sand1990s,Symbolics
madea lisp-based
workstation
anobject-oriented
running
environment.
themachines
weretooexpensive
forthe
programming
Unfortunately,
needsofmostcustomers,
andthecompany
wentbankrupt
itcouldsqueezethe
before
architecture
ontoa chip.)Of the17 seriesfoundbyRamanujan,
thisone converges
thefastest,
givingabout8 digitsofn perterm.AtthetimeofGosper'scalculation,
theworldrecordfordigitsofn wasabout16 milliondigitscalculated
byY. Kanada.
BeforetheBorweinbrothers
hadlaterfounda "conventional"
proofofRamanujan's
series(2.1),theyhadshownthateither(2.1) yieldsan exactformula
forn orthatit
differs
from
n bymorethanIO"3000000.
thathiscalculation
of
Thus,bydemonstrating
n agreedwiththatofKanada,Gospereffectively
hadcompleted
theBorweins'first
proofof (2.1). However,
Gosper'sprimary
goal was notto eclipseKanada'srecord
butto studythe(simple)continued
fraction
ofn forwhichhe calculated
expansion
572
ASSOCIATION
OFAMERICA[Monthly
116
© THEMATHEMATICAL
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Inemailletters
from
terms.
1992andMay,1993,Gosperoffered
17,001,303
February,
thefollowing
remarks
onhiscalculations:
Ofcourse,whatthescribblers
alwayscensoris thatthedigitswerea by-product.
I wanted
tochangetheobjectofthegameawayfrom
decimaldigits.
meaningless
I usedwhatI call a resumable
matrix
towertoexactlycompute
an enormous
to
the
sum
rational
of
a
of
million
terms
of
99~4n
equal
couple
Ramanujan's
=
series.I thendividedtoformthebinary
I
converted
integer floor(7r258'000'000).
toKanadaforcomparison,
thistodecimalandsenta summary
andconverted
the
fraction
to
a
cf
an
fft
based
scheme.
binary
using
1986,D. H. Baileyusedan
Gosper'sworldrecordwas shortlived,as in January,
oftheBorweinsarisingfroma fourth-order
modular
algorithm
equationto compute
ofthecontinued
fraction
of
29,360,000digitsofn. Gosper'scalculation
expansion
mathematical
rewasmotivated
constants
do nothave
bythefactthatmanyimportant
butdo haveinteresting
continued
decimalexpansions
fraction
interesting
expansions.
fraction
areconsiderably
moreinteresting
Thatcontinued
thandecimalexexpansions
brothers
is a viewshared
fraction
[43].Continued
bytheChudnovsky
expanpansions
a constant
beusedtodistinguish
from
whiledecimalexpansions
sionscanoften
others,
thesimplecontinued
fraction
ofe, namely,
likelywillbe unabletodo so.Forexample,
e~
1
1
1
1.
1
1
1
1.
1.
+T + 2 + T + T + 4 + Ï + T + 6 + ! + •••
ofdigitsofe wouldnot
On thecontrary,
has a pattern.
takinga largerandomstring
if
the
continued
fraction
ofn, namely,
e.
It
is
an
one
openproblem simple
help identify
1
111
111iI11J_
+
+
+
7r~3+7
l5 T 292+ ï + ï + T + 2 + T + 3 + ï + Ï4
11111I11_LIII_L
+ 2 + I + T + 2 + 2 + 2 + 2 + T + 84 + 2 + T + T + Ï5
I -
I A. I + 3 + 13+T + 4 + 2 + 6 + •••
hasa pattern.
forn, namely,
seriesrepresentation
a hypergeometric-like
Gosperalsoderived
7=0
'j)Z
whichcan be used to calculateanyparticular
binarydigitof n. See a paperby
and J.Petersson
C. Krattenthaler,
G. Almkvist,
[1] fora proofof (5.1) as well as
theorem.
whichincludethefollowing
generalizations,
k > 1,thereexistsa polynomial
Theorem5.1. Foreachinteger
Sk(j) inj ofdegree
that
such
4kwithrational
coefficients
y^
SkU)
j=Q UfA 4'
2009] ramanujan's seriesfor i/n
August-September
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573
6. JONATHANAND PETER BORWEIN. OnekeytotheworkofbothRamanuforthe
offormulas
for'/n is Clausen'sformula
jan andChowlaintheirderivations
for
the
a
of
the
first
kind
of
or,
by (3.3),
squareof
square
complete
ellipticintegral
thehypergeometric
The
aforementioned
rendition
function
(3.7) of
2^1(5»'' 1; k2).
Clausen'sformula
is notthemostgeneralversion
ofClausen'sformula,
namely,
2F2(a, b'a + b + ''z) = 3F2{2a, 2b,a + b' a + b + '%2a + 2b; z) .
(6.1)
whohaveprovedRamanujan-like
seriesfor1/n
Indeed,theworkof manyauthors
restson specialcasesof(6.1). In particular,
otherhyperultimately
squaresofcertain
ofelliptic
functions
leadonetoRamanujan's
alternative
theories
functions.
geometric
A secondkeystepis to findanother
anformula
for(K/n)2,whichalso contains
otherterminvolving
theterm(K/tt)2
thetwoformulas
toeliminate
l/n. Combining
thenproduces
a hypergeometric-type
seriesrepresentation
for1/tt.Evidently,
unaware
ofChowla'searlierwork,theBorweins
ina similar
fashion
andusedLegproceeded
endre'srelation
[23,p. 24]
E(k)K'(k)+ E'k)K{k) - K(k)K'k) = I
andother
relations
between
toproducesuchformulas.
ellipticintegrals
a seriesrepresentation
fori/n, onenowfacestheproblem
ofevalHavingderived
Ifq = e'71^, then
themoduliandelliptic
thatappearintheformulas.
uating
integrals
forcertain
n onecanevaluatetherequisite
Thisleadsus
positive
integers
quantities.
tothedefinition
oftheRamanujan-Weber
After
classinvariants.
set
Ramanujan,
X(q)'-=(-q'q2)oo,
'q' < 1,
(6.2)
where(a; q)^ is defined
number
andq = e~n^",
rational
by(3.9). If n is a positive
thentheclassinvariants
Gnandgnaredefined
by
Gn:= 2"1V1/24X(<7)
and
gn:= 2-{/4q-l/24X(-q).
(6.3)
In thenotation
of H. Weber[63], Gn = 2"1/4f(vc:ñ)andgn= 2"1/4f1(Vi:n).
As
inSection3,kn:= kie'71^) is calledthesingular
mentioned
In hisvolumimodulus.
nousworkonmodular
setsa := k2andß := I2. Accordingly,
equations,
Ramanujan
we setoin'-k2n.Because[13,p. 124,Entries12(v),(vi)]
- <*)/<?r1/24 and
X(q) = 21/6{<x(l
- a)"2/?}"1724,
X(~q) = 21/6{<*(1
itfollowsfrom(6.3) that
Gn= {4a„(l - a„)}-1/24
and
gn= {4an(l - any2}-i/2'
(6.4)
In theform(6.4),theclassinvariant
sidesof(3.7) and
Gnappearson theright-hand
and
the
values
of
for
several
values
of
n
are
inde(3.10),
Gn
consequently
important
=
certain
series
for
It
is
known
if
that
n
is
and
n
1
riving
1/jr.
(mod4) then
square-free
G4 is a realunitthatgenerates
theHubertclassfieldofthequadratic
fieldQ(V^ñ)
[32,Cor.5.2],andthisfactis veryusefulinevaluating
Gn.Whenwe saythata series
fori/n is associatedwiththeimaginary
field
quadratic Q(v^-ñ), we meanthatthe
constants
involved
intheseriesarerelatedtothegenerators
oftheHubertclassfield
ofQ(V=ïô.
574
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In general,
areactually
numbers.
as
moduliandclassinvariants
algebraic
Singular
seriesfor'/n converges
morerapidly.
Theseries(2.2)
thecorresponding
n increases,
fieldQ('/- 163).
withtheimaginary
is associated
quadratic
seriesforl/n canbe foundintheir
ofall 17ofRamanujan's
TheBorweins'proofs
severalgeneral
seriesreparisefrom
book[23].Theirderivations
hypergeometric-like
andcomplete
ofsingular
classinvariants,
fori/n giveninterms
resentations
moduli,
accountoftheirwork,butwithfewer
de[23,pp. 181-184].Another
integrals
elliptic
ofRamanujan's
thecentenary
tails,canbe foundintheirpaper[24] commemorating
theBorweins
ofRamanujan's
the100thanniversary
Further
birth.
birth,
celebrating
toimaginary
seriesfori/n in[25].Theseriesinthispapercorrespond
further
derived
one
of
their
series
ton = 427
with
with
class
number
fields
2,
corresponding
quadratic
In [29],theauthors
derived
seriesforl/n
about25 digitsofn perterm.
andyielding
ton = 907 yieldfieldswithclassnumber
from
3, witha seriescorresponding
arising
witha field
is
a
series
associated
term.
Their
record
of
n
38
37
or
about
digits per
ing
4 givingabout50 digitsofn perterm;
ofclassnumber
here,n = 1555[28].Thelatter
inthisparagraph.
papergivesthedetailsofwhatwehavewritten
theirworkto a wide
of
havedonean excellent
TheBorweins
job communicating
audience.Besidestheirpaper[28], see theirpaperin thisMonthly [30], with
andtheir
viaworkofRamanujan,
D. H. Bailey,oncomputing
tt,especially
delightful
in
been
which
has
American
in
the
187-199]
[22,
[26],
pp.
reprinted
Scientific
paper
and[11,pp.588-595].
we mentioned
7. DAVID AND GREGORY CHUDNOVSKY. In ourIntroduction
to
calculate
over2 billion
used
and
thattheChudnovsky
brothers,
(2.2)
Gregory David,
of
to
calculate
used(2.2)
1,130,160,664
digits n inthefall
digitsofir.Theyhadfirst
"m zero,"
own
their
then
built
of 1989on a "borrowed"
computer,
They
computer.
of
record
set
a
world
and
in [56],
described
2,260,321,336
digitsofn. The
colorfully
times
since
several
been
broken
n
has
of
worldrecordfordigits
then,andsinceitis
we refrain
from
this
delineate
to
of
this
notthepurpose
history,
computational
paper
records.
further
mentioning
for
theircalculations
examined
haveextensively
TheChudnovskys,
amongothers,
each
In
for
is
normal.
that
n
Itis a longoutstanding
k,
particular,
conjecture
patterns.
0 < k < 9,
Ä
N digitsofn
ofk inthefirst
#ofappearances
Ñ
-
1
TO"
The Chudnovskys'calculations,and all subsequentcalculationsof Kanada, lendcredence to thisconjecture.As a consequence,theaverageof thedigitsovera long in4.5. The Chudnovskysfoundthatforthefirstbillion
tervalshouldbe approximately
digitsthe averagestaysa bit on the high side, while forthe nextbilliondigits,the
statistical
averagehoversa bit on thelow side. Theirpaper [43] gives an interesting
of
a
the
maximal
For
stringof
length
example,
analysisof thedigitsup to one billion.
or
10.
is
either
identicaldigits,foreach of thetendigits,
8, 9,
forl/n9which
deduced(2.2) froma generalseriesrepresentation
The Chudnovskys
=
and
For
r€?Y
definitions.
several
we will describeaftermaking
{r:Imr>0}
is
defined
series
the
Eisenstein
each positiveintegerk,
E2k(x)
by
Ah
~
*2*(t) := 1 - ■=-I>*-iOV>
q = eni'
2009] ramanujan'sseriesfor '/n
August-September
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(7.1)
575
number
andok(n)= Yld'ndk-Klein'sabwhereBk,k > 0, denotesthekthBernoulli
solutemodular
J-invariant
is defined
by
then[13,pp. 126-127,
Itis wellknown
thatif<x(q):= k2(q)9wherek is themodulus,
Entry13]
4(1 -a(q)+a2(q))3
moduli,classinvariants,
Thus,(6.4) and(7.3) showthat,whenq = e~n^, singular
related.
Nowdefine
areintimately
andthemodular
/-invariant
E6(t) V
nlrnrj
We are now readyto statetheChudnovskys'
mainformula[44, p. 122]. If r =
then
+
(1 y=^)/2,
vMl
¿¿16(
J2Wj+
1
^V^JM
] (6m)!
Mj (3/x)!M!31728M/M(r) ^
1
Jn(l-J(r))'
(7.4)
TheChudnovskys'
series(2.2) is thespecialcasen = 163of(7.4).
ideasindirections
TheChudnovsky
andextended
brothers
Ramanujan's
developed
different
fromthoseofotherauthors.
representaTheyobtained
hypergeometric-like
tionsforothertranscendental
forexample,
that
constants
andproved,
Lhl
and
n}24L
r(|)r(i)
aretranscendental
ofthehyperge[42].Theiradvancesinvolvethe"second"solution
ometric
differential
of lineardifferential
equation.Recallfromthetheory
equations
that2F' (a, b; c; x) is a solution
ofa certain
lineardifferential
second-order
equation
witha regular
sosingular
pointattheorigin[5,p. 1]. A secondlinearly
independent
lutionis generally
notanalytic
at theorigin,
andin [43] and [44,pp. 124-126],the
brothers
establish
newhypergeometric
thelatseriesidentities
Chudnovsky
involving
terfunction.
Theiridentities
leadtohypergeometric-like
fortc9
seriesrepresentations
listof such
(5.1). In [45],theauthors
including
Gosper'sformula
providea lengthy
examples,
including
45,+644=¿8J(430J'-«f)J-S20).
TheChudnovsky
brothers
havealsoemployed
seriesfor'/tctoderivetheorems
on
measures
which
are
defined
/z(a)>
irrationality
by
ß(a) := infI 'jl > 0 : 0 < a
I
576
q
- €Q | .
< - hasonlyfinitely
manysolutions
qß
q
J
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Theorem
[6,p. 66],
Thue-Siegel-Roth
Bythefamous
= 1,
fji{a) =2,
> 2,
ifa is rational,
ifa is algebraic
butnotrational,
ifa is transcendental.
canobtainirrationality
measures
forvariousconstants,
theChudnovskys
the
Although
theworldrecord
onetheyobtainforn is notas goodas one wouldlike.Currently,
measureofn is heldbyM. Hata [54],whoprovedthat/¿(7r)<
fortheirrationality
aremuchbetter
forobtaining
measures
for
8.016045 Theirmethods
irrationality
in
their
series
also
a
as
See
such 7T/V640320,
(2.2).
arising
paperby
expressions,
W.Zudilin[64].
8. RAMANUJAN'SCUBIC CLASS INVARIANTAND HIS ALTERNATIVE
howRamanuj
an-Weberclass
THEORIES. In Sections3,4, 6, and7, weemphasized
for
andothers
moduliwereofcentral
andsingular
invariants
importance Ramanujan
inSection1,inparticular,
in
inderiving
seriesfori/n. Wealsostressed
whofollowed
ideaofreplacing
theclassical
remarkable
after(1.5),thatRamanujan's
thediscourse
function
1; x), r = 3,4, 6,leadstonew
2F](', '' 1; jc)by2F' (£, r-y-'
hypergeometric
theories.
On thetopofpage212 in hislostnotebook
alternative
andbeautiful
[60],
=
r
in
which
is
ana3
an
invariant
a
cubic
class
defines
Xn
(i.e.,
(1.5)),
Ramanujan
in
and
defined
Define
invariants
classical
the
of
Gn
(6.3).
gn
Ramanujan-Weber
logue
function
Ramanujan's
'q' < 1,
f(-q)'-=(q'q)oo,
(8.1)
in(3.9),andtheDedekindeta-function
where(a; q)œ is defined
t)(t)
n(T):= ¿*W fj(l - e2^) =: ql/24f(-q),
= 1
(8.2)
7
cubicclassinvariant
andIm x > 0. ThenRamanujan's
whereq = e2niT
knis defined
by
n"
"
3V3V^/W
3V3
/I + iV5A
K- s- )/
'
whereq = e'71^^, i.e.,r = 'iyfnß.
for1/jtin
a generalseriesrepresentation
Chan,Liaw,andTan [34] established
andChudnovskys
oftheBorweins
formulas
tothegeneral
ofknthatis analogous
terms
we first
need
To statethisgeneralformula,
in termsoftheclassicalclassinvariants.
Define
somedefinitions.
1
a*(q)
1 fn(Q)
21q f'Hq3)
'
Thus,whenq = e'71^^ anda* := a*(e-n^),
(8.3) and(8.4) implythat
K
2009] ramanujan'sseriesfor '/n
August-September
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577
In analogywith(3.6), definethemultiplier
m(q) by
2FX(3, 3, 1,0 )
whereß* = a*(qn). We are now readyto statethegeneralrepresentation
of '/n derivedby Chan,Liaw, and Tan [34, p. 102,Theorem4.2].
Theorem8.1. Forn>', let
a*n(l-a*n)dm(a'ß*)
V*
da
a*='-a*n,ß*=a*
bn= l- 2<,
and
tfn=4<(l-<).
Then
We give one example.Let n = 9; thenct$= |. Then, withoutprovidingfurther
details,
whichwas discoveredby Chan,Liaw, and Tan [34, p. 95].
Anothergeneralseriesrepresentation
forl/n in thealternative
theoriesofRamanujan was devisedby Berndtand Chan [19, p. 88, Eq. (5.80)]. We will notstatethisformulaand all therequisitedefinitions,
butletit sufficeto say thattheformulainvolves
Ramanujan'sEisensteinseries P(q), Q(q) = ^(r), and R(q) = E6(r) at theargumentq = -e~n^ and themodular7-invariant,
definedby j(r) = 17287(r), where
7(r) is definedby (7.2). In particular,
3/3+^'
'^~)
= ~27
27(^-D(9Xn2-l)3 '
¡I
a proofof whichcan be foundin [18]. The hypergeometric
termsare of theform
G);G),(I),'
0'!)3
Berndtand Chan used theirgeneralformulato calculatea seriesfor1/ttthatyields
about73 or 74 digitsof n perterm.
578
ASSOCIATIONOF AMERICA [Monthly116
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thatBerndt,
Chan,andLiaw [20]
Lastly,we concludethissectionbyremarking
forl/n thatfallundertheumbrella
havederived
seriesrepresentations
ofRamanujan's
of
functions.
Because
the
is
connected
theory elliptic
quartic
theory intimately
quartic
theirgeneralformulas
withtheclassicaltheory,
[20,p. 144,Theorem
4.1] involvethe
summands.
Notsurprisingly,
thehypergeometric
classicalinvariants
Gnandgnintheir
terms
areoftheform
Bj= w
•
their
is givenby[20,p. 145]
from
Thesimplest
theory
examplearising
(§)■
M*-
9. THE PRESENT AUTHORS AS DISCIPLES OF RAMANUJAN.As menauthor
tocontinue
wereinspired
twoauthors
tionedinSection2, thefirst
bythethird
In theirfirst
ofRamanujan's
thedevelopment
paper[9],BaruahandBerndt
thoughts.
toprove13
ofellipticfunctions
ideasin theclassicaltheory
Ramanujan's
employed
andmanynewonesas well.In [7], theyutilized
of Ramanujan's
originalformulas
fiveofRamanujan's17 formuto establish
cubicandquartictheories
Ramanujan's
las in additionto somenewrepresentations.
by thework
Lastly,in [8], motivated
twoauthors
extended
in Section10 below,thefirst
described
ofJ.Guillera,
briefly
forl/n2.For
seriesrepresentations
ideastoderivehypergeometric-like
Ramanujan's
example,
24
-
oo
/ i 'm+i
= V(44571654400/x2
+ 233588841)BM- + 5588768408/x
,
where
ß (i)
(i)
(L) (i)
(1) (1)
V4/v V4//X-VV2/v '2/ß-v V4/y '4/ß-v
_ Y^
ß~t^
v!3(/¿-v)!3
•
seriesP(q) in(3.12) andoffered
Eisenstein
In Section3, we defined
Ramanujan's
in givinga brief
of ellipticfunctions
theories
fromRamanujan's
severaldefinitions
theroleof P(q) inmoredetail
ideas.Herewe highlight
toRamanujan's
introduction
beforegivinga complete
proofof (1.3). Becausethesethreeseriesrepresentations
ourproofhereis similar
lostnotebook,
(1.2)-(1.4) canalsobe foundinRamanujan's
tothatgivenin[3,Chapter
15].
set
Ramanujan,
Following
z:=2F,(M;l;*).
(9'1}
areRamanujan's
inourderivations
Thetwomostimportant
representation
ingredients
forP(q2) givenby[13,p. 120,Entry
9(iv)]
P(q2) = (1 - 2x)z2+ 6x(l - x)zj-x
2009] ramanujan's seriesfor i/n
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(9.2)
579
intheform
andClausen'sformula
(3.7),which,
using(9.1),we restate
z2 = 3F2(ì, ì, Ì; 1, 1; X) = f^AjX*,
(9.3)
7=0
as in(1.1),
where,
(1)3
and
Aj := ^L
X := 4x(l - x).
(9.4)
From(9.3) and(9.4),
Izj-X
= ¿ A^X^-1-4(1- 2jc).
(9.5)
7=0
Hence,from
(9.2),(9.3),(9.5),and(9.4),
00
00
7=0
7=0
+ 3(!- 2jc)E A^'X;
P(q2)= (1- 2x)J2AJXJ
=
¿{(1
7=0
- 2x) + 3(1 - 2x)j)AjXJ.
(9.6)
Forq := ^"7r^,recall(9.1) andset
xn= kHe-*^),
zn:= 2Fi (¿, ^; 1; x«)
(9.7)
and
(9.8)
Xn=4xn(l-xn).
Forlateruse,we notethat[3,p. 375]
1 - xn= xx/n
and
z'/n= Vnzn.
(9.9)
Withtheuseof(9.7) and(9.8),(9.6) takestheform
P{e-i**) = ¿{(i
7=0
_ 2xn)+ 3(1 - 2xn)j}AjXi
= (1 - 2xn)z2n
+ 3 ¿(1
7=0
- 2xH)jAjXi.
(9.10)
both
eachinvolving
In orderto utilize(9.10),we requiretwodifferent
formulas,
a transformation
Thefirst
comesfrom
P(q2) andP(q2n),wheren is a positive
integer.
forf(-q)
forP(q)> whichin turnarisesfromthetransformation
formula
formula
n. This
in(8.2),andis forgeneral
defined
in(8.1) ortheDedekind
defined
etafunction
transformation
formula
is givenby[13,p. 43,Entry
27(iii)]
f(-e~2ß),
e-a/ì2aì/4f(-e-^) = e-ß/l2ßl/4
580
(9.11)
116
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whereaß = TF2,
witha andß bothpositive.
ofbothsidesof
Takingthelogarithm
we
find
that
(9.11),
°°
--OL + -1 loga +
J>g(l
ñ
- *-2>«)= -£-+
1
oo
logß+ ^log(l - e-2^).
(9.12)
bothsidesof(9.12) withrespect
toa, we deducethat
Differentiating
11^
12 4a
^
1
4a
2je-* = ß
1 - «r2*» 12a
A (2yj8/«)e-^
'
1 - e~2^
^
V' '
bothsidesof(9.13)by 12a andrearranging,
we arrive
at
Multiplying
>-*Efe)=*('-M|:rW)- »i«
. -2ja
'
»
/
• -2#
'
(oo
a = Tt/y/n,
so thatjß = n^/ñ,recalling
thedefinition
(3.12) of P(q), and
Setting
we seethat(9.14)takestheshape
rearranging
slightly,
^
71
= P(e-2«^) + nP(e-2n^).
(9.15)
Thisis thefirst
desiredformula.
forRamanujan's
function
Thesecondgivesrepresentations
[57],[58,pp.33-34]
fn(q):=nP(q2n)-P(q2)
(9.16)
n. (Ramanujan
forcertainpositiveintegers
[57], [58,pp. 33-34] usedthenotation
In
recorded
forfn(q) for12
instead
of
fn(q).) [57],Ramanujan
representations
f(n)
be proved.Theseformulas
ofhowthesemight
valuesofn,buthe gaveno indication
21 ofRamanujan's
inChapter
secondnotebook
arealsorecorded
[59],andproofs
may
be foundin[13].
of
We nowgivethedetailsforourproofof (1.3), whichwas clearlya favorite
inHighSchoolMusical.Unfortunately,
student
we
GabriellaMontez,theprecocious
she possesseda proofof herown.We restate(1.3) herefor
do notknowwhether
convenience.
Theorem9.1. IfAjtj > 0, is defined
by(9.4),then
withmodular
ofdegree7. Thus,our
(9.17)is connected
equations
Proof.Theidentity
modulus
first
taskis tocalculatethesingular
jc7.To thatend,webeginwitha modular
ofdegree7
equation
+ {(1 -x(q))(l
{jc(<7M<77)}1/8
-x(q7))}l/* = 1,
2009] ramanujan'sseriesfor i/n
August-September
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(9.18)
581
inEntry19(i)ofChapter
in 1834butrediscovered
duetoC. Guetzlaff
byRamanujan
of a
19 of his secondnotebook[59], [13,p. 314]. In thenotation
of ourdefinition
=
=
modular
we
and
so
after
have
set
(3.5),
x(q7) I2(q). Set
x(q) k2(q),
equation
=
in
q e-n/yñ (9.18) anduse(9.9) and(9.8) todeducethat
= 1
2 {x7(l - *7)}1/8
and
X7 = 1.
(9.19)
thesingular
notebook
calculated
modulus
x-jinhisfirst
[59],[15,p. 290],
Ramanujan
from
orfrom(9.19),we easilycandeducethat
which,
l-2x7 = ^-.
o
(9.20)
In thenotation
either
5(iii)],
(9.16),from
[57],[58,p. 33],or[13,p. 468,Entry
Mq) = 3z(q)z(q7) (l + y/x(q)x(q1)+ y/(l- x(q))(l - x(q^
.
(9.21)
in(9.21) andemploying
(9.9) and(9.19),we findthat
q = e~n/y/1
Putting
= 3V7
Me'*'^)
(l
+
= 3Vl • 9-z2v
2/c7(l-;c7)) ¿j
(9.22)
n = 7 in(9.15) and(9.10),andusing(9.20),we seethat
Letting
-
n
= P(e~27r/V1)
+ lP(e-2nV1)
(9.23)
and
P(e~2^)
= (1 - 2Xl)z2+ 3
¿(1
- 2x7)jAjXJ7
from(9.22) and(9.23) andputting
theresulting
P{e~2lt/yñ)
respectively.
Eliminating
formula
forP{e-2n^) in(9.24),we findthat
3V7 27V7 2 _ 3V7 2 9^7 ^ jAj
+
In +16.7Z?"
8 Zl ~8~f^0^~'
whichuponsimplification
withtheuseof(9.3) yields(9.17).
■
10. JESÚS GUILLERA. A discrete
function
if
A(n,k) is hypergeometric
A(n + l,k)
A(n,k)
an
A(n,k+1)
A(n,k)
arebothrational
A pairoffunctions
functions.
F(n,k) andG(n,k) is saidtobe a WZ
H. S. WilfandD. Zeilberger
and
pair(after
) ifF andG arehypergeometric
F(n + 1,ik)- F(n, k) = G(n,k + 1) - G(n,it).
582
ASSOCIATION
OFAMERICA[Monthly
116
© THEMATHEMATICAL
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All use subject to JSTOR Terms and Conditions
Inthiscase,H. WilfandD. Zeilberger
existsa rational
function
[55]showedthatthere
such
that
C(nyk)
G(n,k)= C(n,k)F(n,k).
of(F, G). Defining
Thefunction
C(n, k) is calleda certificate
rc+ 1) + G(n,n + k),
H(n, k) = F(n + ifc,
showedthat
WilfandZeilberger
00
00
n=0
n=0
]T#(n,0)= ^G(/i,()).
tousethismethod
to
andZeilberger
Ekhad(Zeilberger's
[46]werethefirst
computer)
of
the
derivea one-page
representation
proof
+
!_g(_,,iWi)§L.
ao.1,
recorded
The identity
(10.1) was first
provedbyG. Bauerin 1859 [10]. Ramanujan
in
second
notebook
his
10
7
of
in
14
Section
as
[59],
[12,
Chapter
(10.1) Example
Bauer's
In
in
found
can
be
references
Further
1905,
[9].
apgeneralizing
pp.23-24].
for
series
further
found
Glaisher
W.
L.
J.
l/n.
[47]
proach,
Motivated
bythiswork,Guillera[48] foundmanynewWZ-pairs(F, G) anddeformurivednewseriesnotonlyfor'/n butforl/n2as well.Oneofhismostelegant
las is
128_ y'
+ 180^+ 13)
1 yY2A5(82°J2
discovered
Guilleraempirically
manyseriesofthetype
Subsequently,
Ä +E
Bj2 + DJ
=
2>
w •
*
A
7=0
itappearsthat
cannotbe provedbytheWZ-method;
Mostoftheserieshe discovered
tothoseseriesfor1/ttwhenH is a powerof2. An
is onlyapplicable
theWZ-method
tobe provedis [49]
exampleofGuillera'sserieswhichremains
128V5
= g(_iy(M^JM|)i(5418y.2+ 693J
+ 29).
seeZudilin'spapers[65],[66].
seriesfor1/7T2,
Forfurther
Ramanujan-like
inthispaperthatClausen's
11. RECENT DEVELOPMENTS. Wehaveemphasized
seriesrepreof
in
most
formula
proofs Ramanujan-type
(6.1) is anessential
ingredient
notdepend
do
that
of
series
for
kinds
other
are
there
forl/n. However,
sentations
l/n
2009] ramanujan'Sseriesfor i/n
August-September
This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:10:04 UTC
All use subject to JSTOR Terms and Conditions
583
uponClausen's formula.One suchseriesdiscoveredby TakeshiSato [62] is givenby
i
3 km
+ '(^-l'Uß •
v
j
U-M^
¿120(4V5-9)=§SUH
^^-^-j
vis
v^v^V^+^V1
(11.1)
In unpublishedwork(personalcommunication
to the thirdauthor),Sato deriveda
morecomplicatedseriesforl/n thatyieldsapproximately
97 digitsof n perterm.A
companionto (11.1), whichwas derivedby a newmethoddevisedbythethirdauthor,
S. H. Chan,and Z.-G. Liu [31], is givenby
jg -£*-&-•
<u-2)
where
We citethreefurther
new seriesarisingfromthisnew method.The firstis another
companionof (1 1.2), whicharisesfromrecentworkof Chan and H. Verrill[36] (after
theworkof Almkvistand Zudilin[2]), and is givenby
9
+ v'(3v)l
f>f ( n-.3u-3v^VM
}
-
UA
i^n'hU
(IV ■
J(v!)3(4/A+1)UJ
The secondis froma paperby Chan and K. P. Loo [35] and takestheform
where
c-elEO'SC:')1)v=0 [j=0 V-»/ <=0 '
'
/ J
The thirdwas derivedby Y. Yang (personalcommunication)
and takestheshape
Motivatedbyhis workwithMahlermeasuresand newtransformation
formulasfor
M.
D.
has
also
discovered
series
for'/n in the
5F4 series,
Rogers[61, Corollary3.2]
of
the
formulas
above.
For
if
is
then
spirit
example, aß definedby (1 1.3),
2
Ä
1V£ 3/x+ l
This serieswas also independently
discoveredby Chan and Verrill[36].
584
© THE MATHEMATICALASSOCIATION OF AMERICA [Monthly 1 16
This content downloaded from 158.135.191.86 on Mon, 09 Mar 2015 18:10:04 UTC
All use subject to JSTOR Terms and Conditions
to W. Zudilin[67], G. Gourevichempirically
discovereda hyperAccording
series
for
l/n3,namely,
geometric-like
forl/nm,m > 2, aredeseriesrepresentations
Thisseriesandthesearchforfurther
ina paperbyD. H. BaileyandJ.M. Borwein[4].
scribed
is thatitshouldgenerate
more
12. CONCLUSION. Onetestof"good"mathematics
concluded
that
Readers
have
undoubtedly
Ramanujan's
original
"good"mathematics.
seriesforl/n havesowntheseedsforan abundant
cropof"good"mathematics.
toR. WilliamGosperforcarefully
ACKNOWLEDGMENTS. The authorsaredeeplygrateful
readingearlier
We are pleasedto thank
excellentsuggestions.
severalerrors,and offering
versionsof thispaper,uncovering
Si Min Chan and Si Ya Chan forwatchingHigh SchoolMusical,thereby
makingtheirfatherawareof Walt
forl/n.
in Ramanujan'sformulas
interest
DisneyProductions'
The researchof Baruahwas partiallysupported
by BOYSCAST FellowshipgrantSR/BY/M-03/05from
DST, Govt.ofIndia.Berndt'sresearchwas partially
bygrantH98230-07-1-0088fromtheNational
supported
of Singapore
SecurityAgency.Lastly,Chan's researchwas partiallysupportedby theNationalUniversity
AcademicResearchFundR-146-000-103-112.
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NAYANDEEP DEKA BARUAH is Readerof MathematicalSciences at TezpurUniversity,
Assam,India.
in 2001. He was a recipientof an ISCA YoungScientist
He receivedhis Ph.D. degreefromthisuniversity
Awardin 2004 and a BOYSCAST Fellowshipin 2005-06 fromtheDepartment
of Science and Technology,
of India,underwhichhe spenttheyearMarch2006-March2007 at theUniversity
of Illinoisat
Government
workingwithProfessorBerndt.He has been a fanof n sincethefirstyearof his B.Sc.
Urbana-Champaign
and becamea hugefanof
To learnmoreaboutn, he beganreadingaboutRamanujanand his mathematics
2000 digitsofn frommemoryin his leisuretime.
Ramanujan!He stilllovesto recitethefirst
Sciences,TezpurUniversity,
Assam,India
Napaam-784028,Sonitpur,
Department
ofMathematical
in
nayan@ tezu.ernet.
of Mathematics
at theUniversity
of Illinoisat Urbana-Champaign.
Since
BRUCE C. BERNDT is Professor
1977he has devotedalmostall ofhisresearchenergytowardfinding
proofsfortheclaimsmadebyRamanujan
and6
inhis(earlier)notebooksandlostnotebook.Aidinghiminthisendeavorhavebeenmanyofhis25 former
visitors.So far,he haspublishedtenbooks
as wellas severaloutstanding
current
doctoralstudents,
postdoctoral
he and GeorgeAndrewshavepublishedtheirsecondof approximately
on Ramanujan'swork.Most recently,
fourvolumeson Ramanujan'slostnotebook.
University
ofIllinois,1409 WestGreenSt.,Urbana,IL 61801, USA
Department
ofMathematics,
edu
berndt@illinois.
of Singapore.He reof Mathematics
at theNationalUniversity
Professor
HENG HUAT CHAN is currently
B. C. Berndt(thesecondauthorofthispaper)and
ofProfessor
ceivedhisPh.D. in 1995 underthesupervision
He thentookup a one-yearvisiting
forAdvancedStudyafterhisgraduation.
spentninemonthsat theInstitute
inTaiwan.Whilein Taiwan,he attempted
without
successto
positionat theNationalChungChengUniversity
designa graduatecoursebasedon thebook"Pi andtheAGM" byJ.M. BorweinandP. B. Borwein.However,
seriesfor1/7T."Since then,he has
it was duringthisperiodthathe beganhis researchon "Ramanujan-type
publishedseveralpaperson thistopicwithvariousmathematicians.
NationalUniversity
ofSingapore,2 ScienceDrive2, Singapore117543,
ofMathematics,
Department
RepublicofSingapore
matchh
@ nus.edu.sg
2009] ramanujan'sseriesfor '/n
August-September
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All use subject to JSTOR Terms and Conditions
587