NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 121-127

UNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS

S. B hargava and Syeda N oor Fathima

(Received April 2003)

Abstract. We obtain a modular transformation for the function

a(q, C, z) = S qn2+nm +m 2cn+m zn -m ) namely,

a (e~2nt, el‘p, eld) = — exp[—( —— ^ )] a(e~ 3t~, e t , e $ i ), t\/3 67vt

and demonstrate how this unifies cubic modular transformations recently found by S. Cooper and their precursors established by J.M. Borwein and P.B. Bor- wein. Here and throughout the paper, unless otherwise stated, it is understood that the summation index or indices range over all integral values.We employ some properties of a(q, £, z) and its relation with the classical theta function (in Ramanujan’s notation) f(a ,b ) = £ a n(n + 1) / 26n(n - 1 )/2 simply es­tablished by the first author in an earlier publication. Also used are some sim­ple properties of f(a ,b ) and a well known modular transformation for f(a ,b) as recorded by Ramanujan.

1 . Introduction

The function

a (q ,( ,z )1 __ ^ gTi2+ n m + m 2^ m + n ^ n —m ( l . i )

\q\ < 1 , C, ^ ^ z was introduced by the first author in [2] and was shown to have properties which unified and generalized several known properties of the Hirschhorn-Garvan-Borwein cubic analogues1 a(q,z), a'(q,z), b(q,z) and c(q,z) of classical theta function [4]:

a(q,z) (1 .2)

a'{q, z) \ s n2 +nm+m2 n•— Z ^ q ’ (1.3)

b(q, z) \ ' „ n 2+nm-\-m2, ,m—n„n■ = / & z » (1.4)

and

c(q,z) := \ ' „n 2+ n m + m 2+ n + m + | ~n—m - z (1.5)

2000 A M S Mathematics Subject Classification: 54B20, 54E15, 54E55.Supported by the UG C Grant No. F 6-129 /2001.xThe functions c(q ,z ) in [4] and c(q, z) in [2] each differ from the ones defined here by means of (1.5) and (1.8), by a factor of q1/ 3.

122 S. BHARGAVA AND SYEDA NOOR FATHIMA

where to = e 2™. That these functions are indeed obtainable from (1.1) can easily by seen by putting £ = 1 in (1 .1) and in the following variants of (1 .1 ).

a'(q, C,z) := a(9>CV^ 1/ 2, r 1/ 2zI/2) = (1.6)

b(q, ( ,z ) := a '(5^ z u , 2) = £ g " 2+"™+mV - " C ” A (1.7)

andc(q, C, z) := 51/ 3a(5, q(, z) = q n2+ n m + m * + n + m + ± ^ n + rr ^ n -m ^ g)

Indeed,

a(q,z) = a{q,l,z), a'(q,z) = a'(q,l,z),

b(q,z) = b(q,l,z) and c(q,z) = c(q,l,z). (1.9)

In his recent paper [5], S. Cooper established the following elegant modular trans­formations for a'(q, z), a(q, z), b(q, z) and c(q, z) which generalize those of J.M. Bor- wein and P.B. Borwein [3] for a(q) := a{q, 1), a'(q) := a'(q, 1), b(q) := b(q, 1) and c(q) c(q, 1):

92af(e~ ,-e ) = ^ expr 6. f a(e 3t 5e 3t)5

27r£ 7

e2 \ eXp|- 6 ^ j

c(e 3t ,e 3t),

and

c(e' “ e,9) = ^ ex p ( - £ ) b{

2n 6e 3t , e ‘ ).

(1.10)

( 1.11)

(1.12)

(1.13)

The main purpose of the present paper is to establish (Section 2) the transfor­mation

a(e~27rt, eiip, ew) =W 3

exp + 30267ft a(e 3* , e ‘ , e^) . (1-14)

Moreover, we show (Section 3) that this is in fact equivalent to the following gen­eralizations of (1.10)-(1.13),

a(e~27rt, eiip,,eid)

a'(e~2nt, ei(p. eie)

b{e~2nt, eilfi,,eie)

t V 3

1ty/ 3

1tV5

exp

ip2 + 3 6>2 Qirt

<p 2 - y e + e 2

67r£

// _2tl 2(£ 39 + >a (e 3‘ , e 3t ,e 3t ), (1-15)

a(e 3‘ , e^ , e 6t<P), (1-16)

v 2 - ipQ + e2 (p 6 nt 31

/ _27t j£_ 20 — veye 3* , e 2t,e et j 9

(1.17)

UNIFICATION OF MODULAR TRANSFORMATIONS 123

and

e x p ( ^ ) c ( e - 2'r,, e - i* , e il>)

= ^ 5 expip2 + 3 e2

67rt

2 ir 2 i£ 30 + ifi .x b[e 3‘ , e 3t ,e at ). (1.18)

That Cooper’s identities (1.10)—(1.13) are special cases of (1.15)—(1.18) is immediate by putting ip = 0 in the latter and using (1.9).

We end the paper by applying (Section 4) the transforms (1.15)—(1.18) to a pair of identities in [2], thereby obtaining new identities involving a'(q, z).

2. The Main Transformation

We first note some simple relations between the function (1.1) and the classical theta function (in Ramanujan’s notation [Eqn. (18.1), Ch. —16,1]),

/(a , b) := ^ V ( " + i ) / 2bn(n-i)/2? \ab \ < l (2.1)

that we will need in the course of this paper. It was indeed shown by the first author in [Eqns. (1.29) and (4.2), 2] that simple series manipulations yield, among other results

a ( q , £ , z ) = g3* 2+ 3AM+M2 £2A +m z h a ^ g3(2A+M)/2 ^ qv /2 ^ ^ . 2 )

(A and /z: integers),

a(q, C, 2) = /(?C2~1,9C'1z)/(93Cz3.«3C“12“3)

+ q ( z f(q2( z - \ C ' z) / f a V . C 1*-3 ) . (2.3)A special case of (2.2), with —A = 1 = /i and with £ changed to q(, that will be of our use is

a{q,q(,z) = C_1 a{q, g_1/2C, ql / 2 z).

Or, what is the same, on using (1.8),

c(q,(,z) = z a(q,q~1 / 2 C,q1 / 2 z). (2.2')

We also need the classical theta function transformation [Entry 20,1]

y /a f(e~ a2+na, e_Q2~na) = en2/ 4\ /^ /(e_/32+in/3, e- ^2_in/?), (2.4)

provided a/3 = 7r and ^ ( a 2) > 0. In fact the following two special cases of (2.4) will be used:

—7rt+ i6 —n t—id\ _ 1 f ^ \/ ( e _7rt+ , e ) = ex p (^ -— J / ( c “ ), (2.5)

andn-* I irt in- / 2 / 7rt 0 2 \ r . 27T + 2S 27T-20,

/ ( e - * + - , e- » ) = ^ 7 e x p ( ¥ - - - — ) / ( - e . , - e . ). (2.6)

Indeed, to obtain (2.5) from (2.4), we put a = y / j , P = and n — To

obtain (2.6) from (2.4), we substitute a = /3 — and n = —

We also need the following simple consequences of the definition (2.1) , noted by Ramanujan (Entries 30(ii)and 30(iii), 31 of [1]):

f(a, b) + f ( - a , -b ) = 2 f(a 3 b,ab3), (2.7)

and

f { a , b ) - f ( - a , - b ) = 2 a f Q , ^ a4 b4 j . (2.8)

Lastly, we need the following identity which easily follows from the definition (1.1),

124 S. BHARGAVA AND SYEDA NOOR FATHIMA

a ( y, - ) = a(q,y ,x ). (2.9)

Indeed, we have

(\ ___ / \ n —m

= ^ 9 n,+nm+m2( ^ ) n+m ( I )

= x in+2my 2m (2.10)

This at once transforms to (2.9) on putting m = i + j and n — —j and using (1.1) again.

We are now in a position to state and prove our master formula.

Theorem 2.1. With a (q X ,z ) defined by (1.1), the transformation formula (1.14) holds.

Proof. We have from (2.3) and repeated use of (2.5)-(2.6)

a fe ~ 2nt e i<p e i6^ _ j ^ e -2nt+i(<p-6) ^e -2Tvt-i(<p-6)^j^e -6nt+i(ip+30) ^e -6nt-i(ip+30)^

_|_ 6 -2 trt+i(<p+0) J e -4nt+i(<p-0) e ~i(ip-0)^ j e ~12nt+i(tp+36) e ~i(ip+3

2V31

where

1 exp v?2 + 3 e26 TTt

{a/3 + a'p') (2.

and

-K+tfi-e Tr-<p + e _7r-H£+3£ _ TT-y-38a = f ( e 2t ; e 2* )? p - / ( e et , e 64 )

/ J., n + ip-e 7r -y + 9 . TT + ip + 36 _ 7r-y -36a = f ( - e 2t , —e 2* )? £ = / ( ~ e 6‘ , - e «* ).

UNIFICATION OF MODULAR TRANSFORMATIONS 125

On using (2.7) and (2.8) repeatedly and the trivial identity 2(a/3 + a'/?') = (a + a')(/3 + (3') + (a — ct'){/3 - /?'), equation (2.11) becomes

> 2 + 302a(e , et(p, e ) =

1 exp

r 2 7 T -e + y

X l/(e * , e

67T t_2n+6- y 2 n + 3 8 + v 2 t t - 3 f l - y

) / ( e st , e 3* )

27T + 2 8 4 7 T -e + y y + 3 f l 47T + y + 3e

+ e 31 / ( e t }C * ) / ( e 3t ,e 3* )]

exp ^2 + 36>267ritV s

x [ / («3<z3, q3 C 1 z - 3 )f (q C .Z -\ q C 1 z)

+ 9C */(96C*3. r 1^ 3) / ( « 2C * "1.C "1*)] (2-12)2 7r . 9 + 1/J 3 6 — 1,

e 3t , £ = e at 5 2 = e etwith qThis is same as the identity

o (e -a,rt,e ^ ,e " ) = ^ = e x py2 + 3fl2

67r£/ 27T cp — 30 . .

a(e~3*,e ar .e « ), (2.13)

on employing (2.3) and the property a(g,£,z) = a(g, C *>2 l ) which is an easy consequence of the definition (1.1). Now, (1.14) follows immediately from (2.13) on using (2.9) with x — , y — e^t. □

3. Equivalent Transformations

Theorem 3.1. The transformation formulas (1.15)—(1.18) are all equivalent to the main transformation (1.14).

Proof. That (1.14) and (1.15) are equivalent follows immediately on using the equivalence of (1.14) with (2.13), the symmetry a'(q,(,z) = a'(q,z X ) (which triv­ially follows from the series form in (1.6)) in the right side of (1.15), and finally the first part of (1.6).

Now, (1.16) follows from (1.15) on putting~Vf and# —

— i(2Q' — (pf)21 ' 6 t'

and then using the trivial identity a'(q, (, z) = a'(q, C_1, z~ x) (which again easily follows from the series form in (1.6)).

To prove (1.17), we have from (1.7)

b(e~27rt,e iip,e ie) = a ' f e - ^ e ^ + ^ e ^ " ^ ) .

Applying (1.16) to this gives, on some simplification

b(e~2irt, eiip, eie)

1 exp1

67T ttV s

x a(q,q~1/2C,q1/2z )

2tt

y2?rY

27T<p + t - + + - r

27r'

126 S. BHARGAVA AND SYEDA NOOR FATHIMA

with q — e~%t, ( = and 2 = e ~ ^ e . However, this at once becomes (1.17) on employing (2.2)'.

Lastly, (1.18) follows from (1.17) in exactly the same fashion as (1.16) from (1.15)but with the substitutions

1 - 2 icp' -i(<p' + 39')t = — , (p = ----- -— and 6 — -------— -------.

31' * 31' 31'We will also need the trivial identity c(q, z) — c(q, £, £_1), which of course followsfrom the series form in (1.8). □

4. New Identities Involving a '(q ,C ,z )

We now apply the results of the previous section to identities (1.31) and (1-32) in [2], namely

2 ct(q, £, z)a(q2, ( 2, z2)

= b(q2 )[b(q, C2, C*3) + C 2, r 1 3)]

+ C2 c(q, (, z)c(q2, C2, z2) + C~2 c(q, C_1, 2)c(g2, C"2, z2) (4.1)and

3z a(q, C, z)f(q3 ( ~ 1 , ( ) f { q z ~ 1 iz )

= / ( 0 C - 1 , C ) / 3 ( g * " \ * )

+ b(q)f(q3 z~3, z3)[3 q C ' f i q 9 C"3, C3) - /(<?C \ C)]- (4-2)We may clarify that we have indeed taken in (4.2) a version of (1.32) of [2] conve­nient for our purpose here. The equivalence is easily established on repeated use of the triple product identity for the theta function (Entry 19, [1]),

/(a , b) = (-a,a&)oo(-M&)oo(aM&)oo (4-3)

and the product form of b(q) (Equation (1.6) of [3])

(q\q)lo fA a\b{q) = v m z ■ (4-4)

where, as usual,OO

(A; 4)00 := J J ( l - A g n).n—0

Theorem 4.1.

2 a'{q,C2 ,Cz)a'{q2 X 2 X z)

= [b(q,(2 Xz)b{q2 , ( 2 ,(z)

+ b (q ,C 2 , C 1 z)b{q2 , C 2 , C 1 z)}

+ c(q) x [(2 c(q2, C3, z) + ( ~ 2 c(q2, C“ 3, z)\ (4.5)

3 a'(q6, C2, Cz)f(q3 C \ q 3 C)f(q9 z - \ q 9 z)

= m q 9 C 3 ,q9C3 ) f 3 (q9 z - \ q 9 z)

+ c(q6 )f(q 3 z - \ q 3 z ) [ f ( q C \ q O - f ( q 9 C 3 ,q9 C3)}- (4-6)

UNIFICATION OF MODULAR TRANSFORMATIONS 127

Proof. Put q = e 2nt, ( = 2 = eie in (4.1) and apply (1.15)—(1.18), including the special case of (1.12) with 6 — 0. Some simplification and change of variables (e_ ^ , e ^ , e ‘ into q, £, z respectively) gives (4.2). Similarly, applying (1.15) and (2.6) repeatedly to (4.2) yields (4.6). We omit the details. □

Rem ark 4.2. By putting ( = 1 in (4.5) we obtain an identity of Cooper ((6.3) of [5]),

a'{q,z)a'(q2 ,z) = b{q,z)b(q2 ,z) + c(q)c(q2 ,z).

Rem ark 4.3. We can recast (4.6) into a form analogous to that of (1.32) of [2],

a'(q2, ( 2 X z)

{-q3C 3 ;q6 )oo{-q3 ( 3 ;q6 )oo(-q3z 1;q6)200(-q3z',q6)lo(q6',q6)6 300

( - t f C - 1 ; q2)oo{-qCi q2)oo(q2; q2) oo

q V H -q z -W U i -q z r fU iM l ( E <n9 " V 3 - E C3V ” 2)

+ ( - 9 C _ 1 ; 9 2 ) o o ( - 9 C ; 9 2 )o o ( - 9 3 z ' 1 ; 9 6 )o o ( - 9 3 « ; 9 6 )o o ( 9 2 ; 9 2 )o o ’One need only make repeated use of (4.3) and the dual of (4.5) (Equation (1.7) of[3])

ciq) = ^ ^ m .\Qi Q) 00

We omit the details.

Acknowledgem ent 1. The authors are grateful to the referee for his valuable suggestions which considerably improved the quality of the paper.

References

1. C. Adiga, B.C. Berndt, S. Bhargava and G.N. Watson, Chapter 16 of Ramanu­jan’s second notebook: Theta functions and q-series, Mem. Am. Math. Soc. 53 no. 315 (1985), American Mathematical Society, Providence, 1985.

2. S. Bhargava, Unification of the cubic analogues of Jacobian theta functions, J. Math. Anal. Appl. 193 no. 2 (1995), 543-558.

3. J.M. Borwein and P.B. Borwein, A Cubic counterpart of Jacobi’s identity and the A G M , Trans. Amer. Math. Soc. 323 no. 2 (1991), 691-701.

4. M.D. Hirschhorn, F.G. Garvan and J.M. Borwein, Cubic analogues of the Ja­cobian theta functions 9(z,q), Canadian J. Math. 45 no. 4 (1993), 673-694.

5. S. Cooper, Cubic theta functions, J. Computational and Applied Math., to appear.

S. BhargavaDepartment of Studies in Mathematics University of Mysore, Manasa Gangotri Mysore-500 006 INDIA

sribhargava@hotmail.com

Syeda Noor FathimaDepartment of Studies in MathematicsUniversity of Mysore, Manasa GangotriMysore-500 006INDIA

srinivasamurthyb@yahoo.com