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RAMANUJAN’S
CLASS
INVARIANTS
AND CUBIC
CONTINUED
FRACTION
3
77, 81, 117> 141, 145, 147, 153, 205, 213; 217, 265, 289, 301: 441, 445, 505, 553> 90,
and 198.
After establishing
calculatcd
26 of Ramanujan’s
invariants
several other class invariants
in [16], [17], and [18], Watson
in four furthcr
papers
[19], [20]: [21], and
[22]. Thc values of n considered by Watson depend upon the class numbers for
positive definite quadratic forms of discriminant -n. In the course of his evaluations,
he determined the class invariants for n = 81 [21], 147 [21], and 289 [22]. Thus, 18
class invariants from the list of 21 bave heretofore not# bcen proved.
In this paper> we shall establish
441, 90, and 198.
prove the remaining
R,amanujan’s
Not#e that for each su&
13 class invariants.
class invariants for n = 117? 153,
n; 9[n. In anothcr
paper
[3], we shall
Our starting point is a relation connecting
gn and ggn, found on page 318 of Ramanujan’s first notebook> but not in his second
notebook.
K. G. Ramanathan
[9], [lO] noticed this relation in the first notebook,
but apparently
he never gave a proof.
notebook:
J. M. and P. B. Borwcin
explicitly,
derived
a formula
Also unaware of its appearance
in the first
[4! pp. 145, 1491, although not stating the results
connc&ng
gr, and ggn,, as well as a formula relating
They
[4,
pp.
145:
1461
also
derive formulas connecting Gsln with GTZ
G,) and Gg?,.
and Ggn, and gslTL with g?, and gg!,. In t,he ncxt section. we use one of Ramanujan’s
modular cquations of degrcc 3 t,o establish t,he aforemcnt,ioned formulas connecting
G,, and GgTZ; and .9?, and gg??. The former formula is net, found in thc notebooks,
but it cari be proved along t,he same lines as thc latter.
Of course, the ideas in this papcr cari be utilizcd to establish ot,hcr class invariants
found by Ramanujan when gin, in particular> for n = 27,45.63,81,225,333,765,18,
126,522, and 630. Undoubtedly,
somc of these proofs would be simpler than previous proofs? for cxample,
for 81; 225> 333; 7G5, 126, 522: and 630. In particular,
Watson’s proofs for n = 333,765,522,
and 630 [17] were based on an “empirical
method.”
In fact: the Borweins [4, pp. 147> 149, 1501 employed the aforcmentioncd
formulas to calculate the invariants GZT: Gsl: Gz2~ T and g522.
Ramanujan’s cubic continued fra&ion G(q), first introduced by him in his second
letter to Hardy [13> p. xxvii]; is dcfined for 1~1< 1 by
y2 + y4
yv3
G(y) := 1
+ +
+ T
+ q
+ . . ..
Chan [6] has proved several elcgant theorems for G(y)> many of which are analogues
of well-known propcrtics satisfied by t,he Rogers-R,amanujan
continued fraction
F(y)
:=
z--1 +ï+T+T+..,
li5
y
y2
y3
In his first letter to Hardy. Ramamljan asserted that F(~-~fi)
“cari be exact,ly
folmd if n be any positive rational quantity.”
Alt~hough this claim is correct, and
the values are algcbraic. it is generally quitc diffîc~ll~~to cvalllate F(emXfi)
for
specific values of n. Ste papcrs of R,amanathan [7], [8]. [9], [lO] and Berndt and
Chan [2] for specific evaluations. In contra& to t,his situation, wc prove in Section
4 general formulas for G(--emnfi)
and G(eerfi)
t,hat arise from
connecting Gr,, and Ggn, and gT2and ggTZ,respectively. WC cmphasize
formulas are known for F(eerfi)
and F(-~-~fi).
the formulas
that no such
4
BRUCE
C. BERNDT.
2. Formulas
Theorem
Proof.
HENG HUAT CHAN, AND LIANG-CHENG
Connecting
ZHANG
GTz and Gg?,, and gn and ggn
1. Let
For brevity;
we set G = GT, throngho~~t, thc proof.
We shall employ Ent,ry 19(xii)
[ll], [l, p. 2311. Let
(2.3)
P = {16a$?(l
in Chapter
19 of Ramanujan’s
Q=
an d
- a!)(1 - /?)}“’
second notebook
(f;;;+;)1’4>
where ,0 bas degree 3. Then
Q+-$++=-$)
(2.4)
=O.
Ntow from (1.5) and (1.7), when q = exp(--~fi)>
G =Y GrI = {4a(1
- ti)}-1’24
GgT, = {4B1
- /?)}-1’24.
an d
Hc!nce, by (2.3); when q = exp(-Tfi),
from (2.4).
(2.5)
(G/GsT)~
Set z = (G9JGj3.
+ (G/&JG
i- 2fi
((GGsJ-~
Th us, (2.5) cari bc rewritten
.x’ -
C2.6)
R,earranging
form
F = (GGgr,)-3
2fiGGz3
and using t,hc not,ation
(z2 -
fiG%
+ 2v+G-%
and Q = (G/GgfTj6.
- (GGA3)
Thus,
= 0.
in the form
+ 1 = 0.
(2.1), we find that we cari recast (2.6) in the
+JI)~ = 2@* - 1) ( G* +)2.
RAMANUcJAN’S
CLASS
INVARIANTS
AND
CUBIC
CONTINUED
FRACTION
5
Since z > l>
or
Noting that CE> 0 and solving for x: WC find that
New,
Tltus, squaring
and expanding,
we find that
and
G4(G8+2G4@=ï+~2-l)
=;
(P+@-z);
-2~,-2dm
(P~-2+14c-i)
+ (Il-
-
+~(~+~~)~~2-1~-2~-2~~
(2.10)
Using (2.9) and (2.10) in (2.7), we deducc
X +=/(P
(2.11)
+ @Y)
(P + @-Y)
(g
t,hat
- 2 + J(J)2 - l)($
(g
- 4 + &2
- l)(p2 - 4)).
Recalling that x = (GgTI/GJ3, wc sec t,hat (2.11) is cquivalent
proof is complete.
We next prove thc aforememioned
notebook.
- 4))
to (2.2),
and SO the
result fomld on page 318 in R,amanujan’s
first
BRUCE
6
Theorem
2.
C. BERNDT,
HENG HUAT CHAN. AND LIANG-CHENG
ZHANG
Let
Tlt,en
YJn
(2,13)
Proof.
,
a
Set 9 = ,9n, throughout
thc proof.
Using (1.7) and (1.4)> wc rewrite
(2.5) in the form
Using (1.7) and (1.4) again; we find that
-(.9/~7?
S&ing
Rccalling
form
x = (9gn/,9)3,
+ (.99~,/.9)~ WC dedllce
the notation
(X2 -
(2.12):
h.9%
2~+(.99~Tz)-~ - 2~&~9gn)~
= 0.
that
we sec that the last cquation
cari be rewritten
in the
- j))2 = 2(J12 + 1) (. 92 1 + JJ2.
It shollld now bc clear that thc rcmainder of t,hc proof is complctcly
th<ritfor Theorem 1, and SO wc omit thc rcst of t,he proof.
analogous
to
The cube roots in (2.2) and (2.13) are not vcry at,tra&ve, and l~s~~allyin applications R,amanujan folmd more appcaling expressions for t,hese cube roots. R,amanujati had an amazing ability for denesting and simplifying radicals: and undoubt,edly
wc do not havc thc insights into radicals t,hat R,amanujan had. Howevcr, it seems
quite likely that in sevcral instances Ramamljan llscd thc following elementary resuit from Carr’s book [5! p. 521. S ince Carr docs not give a proof and sincc hc adds
thc extraneous
here.
hypothesis
Qlat, (a* - !>)1j3 bc a Perfect
cube,
we provide
a proof
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4
P
j
N
w sd
L
2
h
i
(
J = f y
G 117
(
g
h
p
a
d
2
3
a
x
3
C
F
t h c. h
2 23
ea
s
.I
i
114
(
+
aa ,
l
l+
2 g e e o3
n
o. t r
e
e
e[
]
N A.
is
,
v
a
) e
GN
h
J)
l. .
.
v
et h
pf 7
ro
r
cn sc a
n
b og
)
s
e
2
v
2
s
d s
,
)
r n r)
o o b2
R
.
, s
4
ac pn
m
)
,
a
)
e
1
s
e
’
n
=
g2
r
l
0p
vg ui e vi
s
/3
4
T
)2
n2
c
A HN
re
el m w o f rn te
o
o1
3 nF ht W
p
l
ia d
pi r
l ( ne t h f2
c 2
r a
2)
m
2( + )1?
e
r
b
i.
DDT
)
ak
2
la u
- 3( = N (
.
l
)
ua
L
? eW
i e t
2T u >b
c
=
2
ts
.
Te
a
3
1
L
r
63
B
1
Zl,= a, + - 2.
C
n k2
.
n = r1
[ a
p
e
a t
es
NN A E I Z A G
c
(u . -
I
RAU C L H N
CE
i
r
m
c2 m r r e3,
a1
e
ra,
t1
m
=
. h as
f
y
.
2
i
{ d/
o
1
( p,
i
r
v
/B
y
iC h a f
1n n) i g i[
4oR n s er1 r ta
wi
i bs
hcv
s ot t
Rivc
vr h u cai n
I ta
o
r t hv t sn e t
ec o he mh
r
f)
2p
y
e
5
FZ
,
. l
G
c
+
}+
G s2 ei etm f r t ] nc h
ba o
T eo a t 1hm
d a wh ,n t i
eas
o.
aa
e
)w
it
f
2i
t
)c
1
r , o eo n r
ee g l
nn
n
o
~
o
RAMANUJAN’S
CLASS
INVARlANTS
AND CUBIC
CONTINUED
FRACTION
9
We apply Lemma 3 with
a, =
Thus, c = 1. We thcrefore
11+G&
9+Gfi
2
*
need to solve
4x3 - 3z =
TO solvc t,his by inspection,
bx
and
2
it pcrhaps
11+M
2
J
.
is bcst, t,o square both sidcs and set x = t2.
T~E,
i(4i-3)2=
It is not difficlllt to see that 1 = 1+ fi/4.
ll+G&
2
.
Hence, 1 = $ dm,
y = fi/4.
Th us, (3.2) follows> and the proof is complcte.
Alternatively? in the notation (2.19) and (2.21); by (2.22);
and> from (2.lG),
we want to solve
UZ(UP - 3)2 = 22 + 12&
(3.3)
Factoring
in Z [ A/?]: WC dedllce from (2.23) that,
N(,UZ)NP(,UP - 3) = N(22 + 12v5)
= 2z . 13.
Then
zt2 = N(U2 - 3) =: ïV(A + Bd?)
Choose
= A2 - 3B2.
A = 1 = B. We now observe that
ïv(U2) = AJ(4 + fi)
as required.
It is eaily chccked
= 13,
that,, whcn u = dm;
(3.3) holds.
Wc then
readily find that
Q<=;(3++&zz).
Thus, (3.2) has been shown once again.
Theorem 5.
(3.4)
(/?+/y2
G53
Proof. Set 7~= 17 in Thcorcm
paper
[ll],
(/y+/z-q’3.
=
1. By Weber’s
trcatisc
[23, p. 7211: or Ramanujan’s
10
BRUCE
C. BERNDT,
HENG HUAT CHAN? AND LIANG-CHENG
Gk
=
5 + &ï?
4
+ (1+
ZHANG
67J3’2
4fi
;
from (2.1) WC find that
Also,
T~US, from Theorem
1;
b
17+5Jï?+
X
(19+5dïq(l3+5dE)
4
113
13+50+
(19+5dE)(l3+5mJ
+
4
I
However> note that
(19 + 5m)(13
G 153
+ 5dï?‘)
= G72 + MOdï?
= (20 + 4fi?
=
C(bmparing the equality above with (3.4); by (3.5), we sce that it, rcmains to show
that
(3.6)
5+&7+
2
d
19+5mxG6.
which is rather curions indeed. Note that
2
lÏ,
RAMANUcJAN’S
CLASS
INVARIANTS
AND CUBIC
CONTINUED
FRACTION
11
Thus, by (3.6), it suffices to show that
In the notation
Factoring
(2.19)
, we dedllce that
[
N(uZ)N2(u2Wc attempt
WC solve
1
1+&?
2
in Z
and (2.21):
3) = ïV(7 + fi)
= 25.
to salve
*4=iv(uzZ-3)
=: N
A+B
-A2+AB-4B2.
Take A = -1 and B = 1. Then u2 = (5 + m)/2
and N(U~)
calculation shows t,hat (3.8) indccd holds. Lastly, we find that
By (3.5), we conclude
Theorem
= 2. A simple
that (3.7) holds to completfe the proof.
6.
(3.9)
Proof.
We apply Theorem
R,amanujan’s
papcr
G49 = lhxE+
(3.10)
It is easily checked
1 with n, = 49. From Weber’s
[ll].
7l’4
2
that,
.
trcatise
[23? p. 7231, or
12
BRUCE
C. BER.NDT?
After a somewhat
lengthy
HENG
HUAT
calculation~
CHAN,
LIANG-CHENG
ZHANG
WC find t,hat
G*d = 9 + 48
49
AND
* (9 + 3&j71’4
2
2V5
Thus ,
After a mild calculation,
(3.12)
~+dm=9+4fi+2&/lG+Gfi
=9+&?+2&(3+&-)
= (2 + fi)(3&+
Using (3.10)-(3.12)
in Theorem
1; we deduce
2fi)
that
~~~~ =/T/F(2
+ A)I,6
(3.13)
where
A :=&92
+ 72&)(189
=Gd2OlG
Using this calculation
prove that
in (3.13),
+ 72~6)
+ 7G2v?
WC sce from
(3.13)
and (3.9) that it remains
(3.14)
ZZZ--
;
1G + G&‘+
(3 + &‘)G1~271~4 + ;
GI/?+
(3 + v’?)G’~~~‘I~.
to
R
W
A
M
a
AC
L
N
I U L N,
c 3pw
a
TV AAA A CN
e
pi
n
m
lt
S NN
F B D TR
m
yh
SI , I 1
NA
U
CC
3 ET
DI
O
N
a
d
1
A
R ’ CS N
IS U OA
8
1=
1
r * - .i i e a(
z w
c = 1 Sg
1
i
- 3
7
=
(
=
C
+ 3
1
+
2
s n tt i2
69 7
4
2
)
I
w
m
e e sthn.
2 ;
+ .f 1 1
J 2
i
(+ v
+
f G
e
f(
l7 q
c oi a1,
’
i +/
+i v &
G1 2
e
i3
)
) q.
’
ln t5
s
+
’
4
(
2
vg , )
4
3
4
_ (
+ f
4
W
n
v
tC
T
c
h
h
: =
b
o
d
+ f
(
T
T
7
(
m
a
G 7
a3 .
.
( + &
i
(
c
n
g3
V
i
l
t
2
%j
f
4.
y
,
) 2
p1uw
i n. cl
6
r ss
as
= +
a
t
2
b )
1
1
6 l
o W.
nl
ul
p
d
( (+
o ,,
’
5
o c br
) ’
f[ c
e
o
) .p
e
j]
o .2Rb
r
f
l
m
)
p w 7o oo
+
2
4m
) c
r
o 02 nN ht f
J =s h& l N h
I.
s 1doG h
o
.
a
,
1
c
l
1
i
’ )
e
Te
a[
t
_Y10 =
A
f
s
33
f o 1ho
h
=
+ G
4
y (2
n =r 1
p
r
w
+ (( + 8 i
E
3
S
i
u
,990
P
3
1 + 3
C
.
1G
’ (
2r om r 3 ac
1e k
a
,
n
, -r
’
&
.
a I yto . c o a
o1
9 = (
l
6
,w l t t
3
2
s
)
h s l e
)
+
o
a
1
l
t
&
a
’
BRUCE
14
C. BERNDT,
HENG
T~US, from
(2.13) and (3.16)>
(3.17)
.!J90 =
(2 +
Comparing
(3.17)
and (3.15);
HUAT
CHAN,
{
dwYJs+
AND
LIANG-CHENG
pyz+
ZHANG
/yz}1’3
d%p6
we sec t,hat, wc ml&
prove that
(3.18)
In thc notation
(2.19) and (2.21),
U2(U2 - 3)2 = 18 + &.
(3.19)
Factoring
we salve
in Z[&],
we find that
N(U2)N2(UZ
- 3) = N(18 + GV@ = 3. 62
Wc thus want t,o solvc
&6 = N(U* - 3) =: N(A
+ I?d@ = A* - 6B2
Cho~se A = 0 and B = 1: SO that u,* = 3 + fi
that (3.19) is satisficd. Then
and N(U*)
= 3. It is trivial to see
_/q+#
and thc verification
Theorem
(3.20)
of (3.18) is complctc.
8.
gl‘J8 = 43
(4&+
Proof. Set# n, = 22 in Theorem
paper [ll],
(3.21)
An cas-y calculation
&?)1’6
(p-z+p-q.
2. From Wcber’s
g22
yields p = 4fi.
=
work [23, p. 7221, or R,amanujan’s
1+lh.
Thlls, from Thcorcm
2,
RAMANUtJAN’S
By (3.20)
CLASS
and (3.22),
INVARIANTS
AND CUBIC
CONTINUED
FRACTION
15
we must show that
(3.23)
In t,hc not&ion
of (2.19)
and (2.21)> WC seek to salve
UZ(U2 - 3)2 = 4(18 + 36).
(3.24)
T~US,
,
fV(U2)AqU2
- 3) = ïV(4(18 + 3m))
we attempt
Let A = 1 = B, SO t,hat u* = (9 + a)/2
(3.24) holds.
= 24 . 33
t,o find a solution
of
and N(U~) = 12. It is easily checked that
Hcncc;
*,+jE+/~.
Thus, we cari deduce
4. Explicit
Proof.
(3.23),
and the proof is complctc.
Evaluations
of Ramanujan’s
It was first proved by Watson
Cubic
Continued
Fraction
[15] that
(4.1)
where x(q)
is defined by (1.4).
34G, 3471.) For q = -C
G(
(4.2)
-e-
xfi,
(F or rcferenccs
t,o o&r
WC havc, by (4.1) and (1.7).
proofs of (4.1), sec 11, pp.
1
B
6
C B
R
H.
u
E HX
J
:
CE
RAU J L H N
- 4 + &
(
=
-? l
f
(
a
B
f
(
r(
2w
u
(
f n
r
t o4
2
t
4
S
( u
W
n
i
(
.
l
x
4
9 w
L
S
[
f
E
a
2
P
e
c t
6
p
x
F
o
e
.
Wr
tr
h
G
]r
r
e[o
o
f
tt a
G o+ G
h4
et e ~ x
: al
2
3
t
-
(
,
GN
,
,
e)
oi p
o w s a
o
R
b2o
h
t
r p ca3f
l
+
’
aii
n
l
to l
t m
h
a1
1
r
p
m
-
Ui ,
e
f
a rm
,.
uf c
e
i
e
c
l
[
.
e
~ v 9d y . h ?
o
e2m
r
f
-
p
.
)
l
op = cI2 T i n b f T n.
A
=
h= s f
o
NA,
$
t)
p
o
p r7o
d
rt mo
m
C
I e
l
a
G
.
i
2.
=t 1
r a
T
(
)
m
o.
n
)
8
t . Cps o t
=
)?
)
a2
DDT E A H N
)
. a
n = r1 T
n
m
hxa f h c
G
P
2
,
)
7
4
4W b c n
e Tc
1
o
.
i
E
. e i d hm
.
- 4J
p
2
i
NNA C I Z A G
Sa
o
.
r
e
1
t1
p’ a
i
es n
d
1
F
S
, = (
’O
tl:
3
od
A
4
y
)
w
-)
s
(
&
4
S
4 a
e
l
t
v
it
, Gn
v
v
=h (
l o G 1m
h g a
as ft
b
3 ic i l
t io , t e; lp v u o n
Tfh h gar c e f gh e e iro n
a
6 sd [c & pl 7 J T [ 2 w
? .uc 2 d1 l 3 t+
e eaf 2 ci l
, h or 1 d’ ]
e
a
l
u
a
t
nl o
il , u ;
i
e
s
RAMANUcJAN’S
E
3
x
4
P
INVARIANTS
.
.
ra i
tr p
u
t
?
(
ah
(
G
H
(
W
g
E
L
S
U
f
i
[
f
(
v
l
e
2 ~
( s
o
i o
t
x
6
a3
a
o
t
5
,
n m
h
a1
7r m
e
h
hc
[t [
sv
]
r
i
c
. n. = h ( .a n +9 5d
l
4
l
l
u
s
e
m
c
) r
n)
o l
, .
~
p.
g.
o
l
h
; d c)
i
e
e i 1uat
w
r
s
c
e
. , 1 n n] y,
&
f c
~
l
ew f 2cl g a S )G d
o
,
+
)o p
m0
,
)
y
e
c
2
2 Y
2c
J
9 = f0 1. a
n e l b3 TI2
o&
d f mg7 d 1a
?
)’
p
, .=
o 21 c H l 2
u
3
. 1+
iG
l
j
)
)
t.
v T owl
1
1v
7
n o on. c h
.
J)
)
9 - - Ym
q
I . e.
e.p o
a o r(i1 c 4t d
.
2 ) J
m
3v i
l
. y
( (m
a
c ( 1 po. 7 h
lc
dn
+ (4
p
2
.
(d
=
r efu e C
( ef
+
? 1 (
& e
n t4
,
te
+ 2+ J
(
=
, d
tc i
x
~ = r2 T
a
p
J >2w0
5
P
m
s r
ee
i
a
(o
a f
f
3
(
w4 c
2
+
= -
i u(
-
e
i
ne
.
l
%
d
_2
17
V
l
4
4
P
i
4
f
(
( .
FRACTION
p
o s n ph g r W do
:
CONTINUED
m
&
x
A
AND CUBIC
a
dE)_
G(-em
E
CLASS
c
e
o
l s/ ) e
s
)crh
i
s 2, d
.
,i e
n
o
18
BR,UCE
Example
6.
Example
7.
Example
8.
Example
C. BER,NDT,
HENG
HUAT
CHAN.
6 appcars in Ramanujan’s
AND
LIANG-CHENG
lest, notebook
ZHANG
[14, p. 3G6] and was first
provcd by K.G. Ramanathan [7].
We give our thanks t,o Deug Bowman for informing us of th
3 in Carr’s book [5].
version of Lemma
R,EFERENCES
1. B. C. Berndt:
Nokbooks,
Rumanujan’s
Part
III. Springer-Verlag:
2. B. C. Berndt, and H. H. Chan, Some values for
New York,
1994.
the Ro~ers-Ra?nan,7L~~Lr~ continued
fruction
(to
appear).
3. B. C. Berndt,
formula,
H. H. Chan. and L.-C.
and modular
equations
Pi a&
4. J. M. and P. B. Borwein,
5. G. S. Carr,
Formulas
G. H. H. Chan,
Ramanujan’s
Zhang;
the AGM,
and Th,eorems
On Ramanujan’s
Wilcy,
in Pure
cubic
On Ramanujan’s
On th,e Ro,qers-Ramanujan
Sci.) 93 (1984),
9. K. G. R,amanathan,
fraction,
continued
7. K. G. R,amanathan.
New York.
Mathematics,
continued
8. K. G. R,amanathan,
(Math.
cZass in,variants,
Kronecker’s
limit
(to appear).
1987.
2n,d ed.; Chclsea,
New York,
1970.
(to appear).
fraction,
Acta
continued
Arith.
fraction,.
43 (1984),
209-226.
Proc.
Indian
Acad.
Sci.
J. Pure Appl.
Math.
16 (1985),
67-77.
continued
Ramanujan’s
fraction,
Indian
695-724.
10. K. G. R,amanathan,
(1987),
Some
of Kronecker’s
applications
limit formula,
J. Indian
Math.
Soc. 52
71-89.
11. S. Ramanujan,
Modulur
(1914), 350-372.
Notebooks
12. S. R,amannjan,
equutions
a&
to x.
approzimaiions
(2 ,volumes),
Tat,a Institnt,e
Qnart.
J. Math.
of Fnndamcntal
(Oxford)
R,esearch;
45
Bombay,
1957.
13. S. R,amanujan,
Collected
Papers.
14. S. R,amanujan,
TlAe Lest
Nokbook
15. G. N. Watson,
Theorems
Soc. 4 (1929),
231-237.
1G. G. N. Watson,
Soc. 6 (1931),
TlLeorems
stated
Chclsca.
New York.
by Raman,ujan
stated
19G2.
anci OtlLer Unpuhlisl~,ed Papers.
(IX):
hy Ramanujan
Narosa,
two con,tinued fractions.
(XIV):
a sinqular
modt~h~,
New
Delhi,
Math.
London
Math.
J.
12G-132.
17. G. N. Watson,
Som,e singular
moduli
(I),
18. G. N. Watson;
Some
moduli
(II),
singular
Q nart. J. Math. 3 (1932): 81-98.
Q uart. J. Math. 3 (1932); 189-212.
19. G. N. Watson,
Singular
moduli
(3);
Proc.
20. G. N. Watson,
Singular
moduli
(4).
Acta
(5).
Proc.
London
Math.
Soc. 42 (1937),
377-397.
(G). Proc.
London
Math.
Soc. 42 (1937),
398-409.
21. G. N. Watson,
Sin,qulaT moduli
22. G. N. Watson.
Singular
moduli
London
Arith.
Math.
Soc. 40 (1936),
1988.
J. London
83-142.
140 (193G), 284-323.
R
2
H W
AC
I
Le!~,rbuclt,
3
.
eder Al,qebTa.
.
dritter
b
DEPARTMENT OF MATHEMATE~,
URBANA,IL 61801, USA
MO
ML
AN
Band,
1409 WEST
C AA
C
e
C NV
N
Y h
U NS F O DA
1 r
e
o c
B U
S R
1 N R
9,
w
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DEPARTMENT OF MATIIEMATICS, SOCITIIWESTMISSOURI STATE UNIVERSITY,SPRINGFIELD,
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r l
I
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9A T I
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