... relation with the classical theta function (in Ramanujan's notation) ..... [3] J. M. Borwein and P. B. Borwein, A Cubic counterpart of Jacobi's identity and the AGM ...
UNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS S. Bhargava* and Syeda Noor Fathima Department of Studies in Mathematics University of Mysore, Manasa Gangotri Mysore-500 006 INDIA
Abstract: We obtain a modular transformation for the function a(q, ζ, z) =
P
2 +nm+m2
qn
a(e−2πt , eiϕ , eiθ ) =
ζ n+m z n−m , namely,
ϕ 2 +3θ 2 θ 2π 1 √ exp[−( ϕ 6πt )] a(e− 3t , e t , e 3t ), t 3
and demonstrate how this unifies cubic modular transformations recently found by S. Cooper and their precursors established by J. M. Borwein and P. B. Borwein. 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, ζ, z)and its relation with the classical theta function (in Ramanujan’s notation) f (a, b) =
P
an(n+1)/2 bn(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. *
Supported by UGC Grant No. F6-129/2001/SA-II
2
1. INTRODUCTION The function a(q, ζ, z) =
P
2 +nm+m2
qn
ζ m+n z n−m ,
(1.1)
|q| < 1, ζ 6= 0 6= 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 analogues* a(q, z), a0 (q, z), b(q, z) and c(q, z) of classical theta function [4]: 2 +nm+m2
qn
a(q, z) :=
P
a0 (q, z) :=
P
b(q, z) :=
z n−m ,
2 +nm+m2
qn
2 +nm+m2
(1.2)
zn,
(1.3)
P
qn
ω m−n z n ,
P
qn
2πi 3
. That these functions are indeed obtainable from (1.1) can
(1.4)
and c(q, z) := where ω = e
2 +nm+m2 +n+m+ 1 3
z n−m ,
(1.5)
easily by seen by putting ζ = 1 in (1.1) and in the following variants of (1.1).
a0 (q, ζ, z) := a(q, ζ 1/2 z 1/2 , ζ −1/2 z 1/2 ) = b(q, ζ, z) := a0 (q, ζω, zω 2 ) =
P
2 +nm+m2
P
qn
P
qn
2 +nm+m2
qn
ζ mzn,
ω m−n ζ m z n ,
(1.6) (1.7)
and c(q, ζ, z) := q 1/3 a(q, qζ, z) =
2 +nm+m2 +n+m+ 1 3
ζ n+m z n−m . (1.8)
*
The functions c(q, z) in [4] and c(q, ζ, z) in [2] each differ from the ones defined here by means of (1.5) and (1.8), by a factor of q 1/3 .
3 Indeed, a(q, z) = a(q, 1, z), a0 (q, z) = a0 (q, 1, z), b(q, z) = b(q, 1, z) and c(q, z) = c(q, 1, z).
(1.9)
In his recent paper [5], S. Cooper established the following elegant modular transformations for a0 (q, z), a(q, z), b(q, z) and c(q, z) which generalize those of J. M. Borwein and P. B. Borwein [3] for a(q) := a(q, 1), a0 (q) := a0 (q, 1), b(q) := b(q, 1) and c(q) := c(q, 1): a0 (e−2πt , eiθ ) =
2
2π
θ
2π
θ
θ exp(− 6πt ) a(e− 3t , e 3t ),
1 √ t 3
2
a(e−2πt , eiθ ) =
1 √ t 3
θ ) a0 (e− 3t , e t ), exp(− 2πt
b(e−2πt , eiθ ) =
1 √ t 3
θ exp(− 6πt ) c(e− 3t , e 3t ),
1 √ t 3
θ exp(− 2πt ) b(e− 3t , e t ),
2
2π
θ
2
2π
θ
(1.10) (1.11) (1.12)
and c(e−2πt , eiθ ) =
(1.13)
The main purpose of the present paper is to establish (Sec.2) the transformation a(e−2πt , eiϕ , eiθ ) =
2
2
2π
θ
ϕ
+3θ exp[−( ϕ 6πt )] a(e− 3t , e t , e 3t ).
1 √ t 3
(1.14)
Moreover, we show (Sec.3) that this is in fact equivalent to the following generalizations of (1.10)-(1.13),
a(e−2πt , eiϕ , eiθ ) = a0 (e−2πt , eiϕ , eiθ ) = b(e−2πt , eiϕ , eiθ ) = and
2
1 √ t 3 1 √ t 3
2
2ϕ
2π
+3θ exp[−( ϕ 6πt )] a0 (e− 3t , e 3t , e
1 √ t 3
exp[−( ϕ
exp[−( ϕ
2 −ϕθ+θ 2
6πt
2 −ϕθ+θ 2
6πt
2π
ϕ
3θ+φ 3t
)] a(e− 3t , e 2t , e
)+
),
2θ−ϕ 6t
(1.15) ),
ϕ 2θ−ϕ 2π ϕ ] c(e− 3t , e 2t , e 6t ), 3t
(1.16) (1.17)
4 exp( −2iϕ ) c(e−2πt , e−iϕ , eiθ ) = 3
1 √ t 3 2π
2
2
+3θ exp[−( ϕ 6πt )] × 2ϕ
b(e− 3t , e 3t , e
3θ+ϕ 3t
). (1.18)
That Cooper’s identities (1.10)-(1.13) are special cases of (1.15)-(1.18) is immediate by putting ϕ = 0 in the latter and using (1.9). We end the paper by applying (Sec. 4) the transforms (1.15)-(1.18) to a pair of identities in [2], thereby obtaining new identities involving a0 (q, ζ, 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]), f (a, b) :=
P
an(n+1)/2 bn(n−1)/2 ,
|ab| < 1
(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 2 +3λµ+µ2
a(q, ζ, z) = q 3λ
ζ 2λ+µ z µ a(q, q 3(2λ+µ)/2 ζ, q µ/2 z),
(2.2)
(λ and µ: integers), a(q, ζ, z) = f (qζz −1 , qζ −1 z)f (q 3 ζz 3 , q 3 ζ −1 z −3 ) +q ζ z f (q 2 ζz −1 , ζ −1 z) f (q 6 ζz 3 , ζ −1 z −3 ) . (2.3)
5 A special case of (2.2), with −λ = 1 = µ and with ζ changed to qζ, that will be of our use is a(q, qζ, z) = ζ −1 z a(q, q −1/2 ζ, q 1/2 z). Or, what is the same, on using (1.8), c(q, ζ, z) = q 1/3 ζ −1 z a(q, q −1/2 ζ, q 1/2 z).
(2.2)0
We also need the classical theta function transformation [Entry 20, 1] √
αf (e−α
2 +nα
, e−α
2 −nα
2 /4
) = en
√
βf (e−β
2 +inβ
, e−β
2 −inβ
),
(2.4)
provided αβ = π and Re(α2 ) > 0. In fact the following two special cases of (2.4)will be used: f (e−πt+iθ , e−πt−iθ ) =
1 √ t
2
θ exp(− 4πt ) f (e−
π+θ t
, e−
π−θ t
),
(2.5)
2π−2θ 2π+2θ θ2 ) f (−e− t , −e− t ). 2πt
(2.6)
and f (e−πt+iθ , e−iθ ) =
q
2 t
exp( πt − 8
iθ 2
−
Indeed, to obtain (2.5) from (2.4), we put α = β = α=
q
√
πt, and n =
πt , 2
β=
q
2π , t
√θ . πt
q
π , t
To obtain (2.6) from (2.4), we substitute q
and n = −
πt 2
+i
q
2 πt
θ.
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) = 2f (a3 b, ab3 ),
(2.7)
and b a f (a, b) − f (−a, −b) = 2af ( , a4 b4 ). a b
(2.8)
Lastly, we need the following identity which easily follows from the definition (1.1),
6 a(q, x3 y, xy ) = a(q, y 2 , x2 ).
(2.9)
Indeed, we have a(q, x3 y, xy ) =
P
2 +nm+m2
qn
=
(x3 y)n+m ( xy )n−m
X
2 +nm+m2
qn
x4n+2m y 2m .
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, ζ, 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(e−2πt , eiϕ , eiθ ) = f (e−2πt+i(ϕ−θ) , e−2πt−i(ϕ−θ) )f (e−6πt+i(ϕ+3θ) , e−6πt−i(ϕ+3θ) )+ e−2πt+i(ϕ+θ) f (e−4πt+i(ϕ−θ) , e−i(ϕ−θ) )f (e−12πt+i(ϕ+3θ) , e−i(ϕ+3θ) ) ϕ2 + 3θ2 1 = √ exp[−( )](αβ + α0 β 0 ) 6πt 2 3t (2.10) where α = f (e−
π+ϕ−θ 2t
, e−
π−ϕ+θ 2t
),
β = f (e−
π+ϕ+3θ 6t
, e−
π−ϕ−3θ 6t
)
7 and α0 = f (−e−
π+ϕ−θ 2t
, −e−
π−ϕ+θ 2t
), β 0 = f (−e−
π+ϕ+3θ 6t
, −e−
π−ϕ−3θ 6t
).
On using (2.7) and (2.8) repeatedly and the trivial identity 2(αβ + α0 β 0 ) = (α + α0 )(β + β 0 ) + (α − α0 )(β − β 0 ), equation (2.10) becomes a(e−2πt , eiϕ , eiθ ) =
√1 3t
[f (e− +e−
2
2π−θ+ϕ t
2π+2θ 3t
2
+3θ )]× exp[−( ϕ 6πt
, e−
f (e
ϕ−θ t
2π+θ−ϕ t
, e−
)f (e−
4π−θ+ϕ t
2π+3θ+ϕ 3t
, e−
2π−3θ−ϕ 3t
)
ϕ+3θ 3t
, e−
4π+ϕ+3θ 3t
)]
)f (e
1 ϕ2 + 3θ2 = √ exp[−( )]× 6πt t 3 [f (q 3 ζz 3 , q 3 ζ −1 z −3 )f (qζz −1 , qζ −1 z) +qζzf (q 6 ζz 3 , ζ −1 z −3 )f (q 2 ζz −1 , ζ −1 z)] 2π
with q = e− 3t , ζ = e−
θ+ϕ 2t
,z = e
3θ−ϕ 6t
.
This is same as the identity a(e−2πt , eiϕ , eiθ ) =
ϕ+θ ϕ−3θ 2 +3θ 2 2π 1 √ exp[−( ϕ 6πt )]a(e− 3t , e 2t , e 6t ), t 3
(2.11)
on employing (2.3) and the property a(q, ζ, z) = a(q, ζ −1 , z −1 ) which is an easy consequence of the definition (1.1). Now, (1.14) follows immediately from ϕ
θ
(2.11) on using (2.9) with x = e 6t , y = e 2t .
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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 (2.11), the symmetry a0 (q, ζ, z) = a0 (q, z, ζ) (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 t=
1 , 3t0
ϕ=
−iϕ0 2t0
and θ =
−i(2θ0 −ϕ0 ) 6t0
and then using the trivial identity a0 (q, ζ, z) = a0 (q, ζ −1 , z −1 ) (which again easily follows from the series form in (1.6)). To prove (1.17), we have from (1.7) 2π
2π
b(e−2πt , eiϕ , eiθ ) = a0 (e−2πt , ei(ϕ+ 3 ) , ei(θ− 3 ) ). Applying (1.16) to this gives, on some simplification b(e−2πt , eiϕ , eiθ ) = 1 1 2π 2 2π 2π 2π 2 √ exp[− {(θ − ) − (θ − )(ϕ + ) + (ϕ + ) }]× 6πt 3 3 3 3 t 3 a(q, q −1/2 ζ, q 1/2 z) 2π
with q = e− 3t ,
ϕ
ζ = e 2t
and
z=e
2θ−ϕ 6t
.
However, this at once becomes (1.17) on employing (2.2)0 .
9 Lastly, (1.18) follows from (1.17) in exactly the same fashion as (1.16) from (1.15) but with the substitutions t=
1 , 3t0
ϕ=
−2iϕ0 3t0
and
θ=
−i(ϕ0 +3θ0 ) . 3t0
We will also need the trivial identity c(q, ζ, z) = c(q, ζ, z −1 ), which of course follows from the series form in (1.8).
4. NEW IDENTITIES INVOLVING a0 (q, ζ, z) We now apply the results of the previous section to identities (1.31) and (1.32) in [2], namely 2 a(q, ζ, z)a(q 2 , ζ 2 , z 2 ) = b(q 2 )[b(q, ζ 2 , ζz 3 ) + b(q, ζ −2 , ζ −1 z 3 )] +ζ 2 c(q, ζ, z)c(q 2 , ζ 2 , z 2 ) + ζ −2 c(q, ζ −1 , z)c(q 2 , ζ −2 , z 2 )
(4.1)
and 3z a(q, ζ, z)f (q 3 ζ −1 , ζ)f (qz −1 , z) = f (qζ −1 , ζ)f 3 (qz −1 , z)+ b(q)f (q 3 z −3 , z 3 )[3 qζ −1 f (q 9 ζ −3 , ζ 3 ) − f (qζ −1 , ζ)]. (4.2) We may clarify that we have in deed 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]), f (a, b) = (−a, ab)∞ (−b; ab)∞ (ab; ab)∞
(4.3)
10 and the product form of b(q) (Equation (1.6) of [3]) b(q) =
(q; q)3∞ , (q 3 ; q 3 )∞
(4.4)
where, as usual, (λ; q)∞ :=
∞ Y
(1 − λq n ).
n=0
Theorem 4.1. 2a0 (q, ζ 2 , ζz)a0 (q 2 , ζ 2 , ζz) = [b(q, ζ 2 , ζz)b(q 2 , ζ 2 , ζz)+ b(q, ζ −2 , ζ −1 z)b(q 2 , ζ −2 , ζ −1 z)] + c(q)× [ζ 2 c(q 2 , ζ 3 , z) + ζ −2 c(q 2 , ζ −3 , z)] (4.5) 3a0 (q 6 , ζ 2 , ζz)f (q 3 ζ −1 , q 3 ζ)f (q 9 z −1 , q 9 z) = 3f (q 9 ζ −3 , q 9 ζ 3 )f 3 (q 9 z −1 , q 9 z) + c(q 6 )f (q 3 z −1 , q 3 z) [f (qζ −1 , qζ) − f (q 9 ζ −3 , q 9 ζ 3 )]. (4.6)
Proof. Put q = e−2πt , ζ = eiϕ , z = eiθ in (4.1) and apply (1.15)-(1.18), including the special case of (1.12) with θ = 0. Some simplification and change π
ϕ
θ
of variables (e− 3t , e 3t , e t into q, ζ, z respectively) gives (4.2). Similarly, applying (1.15) and (2.6) repeatedly to (4.2) yields (4.6). We omit the details.
11 Remark 4.1. By putting ζ = 1 in (4.5) we obtain an identity of Cooper ((6.3) of [5]), a0 (q, z)a0 (q 2 , z) = b(q, z)b(q 2 , z) + c(q)c(q 2 , z). Remark 4.2. We can recast (4.6) into a form analogous to that of (1.32) of [2], a0 (q 2 , ζ 2 , ζz) = (−q 3 ζ −3 ; q 6 )∞ (−q 3 ζ 3 ; q 6 )∞ (−q 3 z −1 ; q 6 )2∞ (−q 3 z; q 6 )2∞ (q 6 ; q 6 )3∞ (−qζ −1 ; q 2 )∞ (−qζ; q 2 )∞ (q 2 ; q 2 )∞ 2
2
q 2/3 (−qz −1 ; q 2 )∞ (−qz; q 2 )∞ (q 6 ; q 6 )2∞ ( ζ n q n /3 − ζ 3n q 3n ) . + (−qζ −1 ; q 2 )∞ (−qζ; q 2 )∞ (−q 3 z −1 ; q 6 )∞ (−q 3 z; q 6 )∞ (q 2 ; q 2 )∞ P
P
One need only make repeated use of (4.3) and the dual of (4.5) (Equation (1.7) of [3]) c(q) = 3q 1/3
(q 3 ; q 3 )3∞ . (q; q)∞
We omit the details.
ACKNOWLEDGEMENT
The first author S. Bhargava is thankful to University Grant Commission, Government of India, New Delhi for the financial support under the grant F6-129/2001/SA-II. The authors are grateful to the referee for his valuable suggestions which considerably improved the quality of the paper.
12
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(1985), no. 315, American Mathematical Society, Providence, 1985. [2] S. Bhargava, Unification of the cubic analogues of Jacobian theta functions, J. Math. Anal. Appl. 193(1995), no.2, 543-558. [3] J. M. Borwein and P. B. Borwein, A Cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc. 323(1991), no.2, 691-701. [4] M. D. Hrischhorn, F. G. Garvan and J. M. Borwein, Cubic analogues of the Jacobian theta functions θ(z, q), Canadian J. Math. 45(1993), no.4, 673-694. [5] S. Cooper, Cubic theta functions, preprint.
