THE BORWEINS' CUBIC THETA FUNCTIONS AND q-ELLIPTIC

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In section 2 we will apply Lemma 1 to a certain q-elliptic function and thereby establish (Theorem 2) an old, very general, identity originally stated by Weierstrass.
THE BORWEINS' CUBIC THETA FUNCTIONS AND q-ELLIPTIC FUNCTIONS Richard Lewis SMS, The University of Sussex, Brighton BNl 9QH, United Kingdom r.pJewisC)susx.ac.uk

Zhi-Guo Liu Mathematics Dept., Xinxiang Education College, Xinxiang, Henan 453000, P.R. China, and Nanjing Institute of Meteorology, Nanjing 210044, P.R. China liuzg18c)hotmail.com

Abstract

We give two identities; one, originally due to Weierstrass, is old and one appears to be new. We use these identities to give proofs of some relations between the functions a(q), b(q) and c(q) of Borwein, Borwein and Garvan

[4].

Keywords: q-elliptic function, the power series tities

1.

(Zj

q)oo and [z; q]oo, the Borweins' iden-

INTRODUCTION

Suppose throughout that q is a complex number of modulus < 1 (we will occasionally suppose that q = exp(27riT), where T has positive imaginary part) and set 27ri

w:=e 3

.

We will use the familiar notation

II (1 00

(z; q)oo :=

zqn)

n:::O

and we also set

133 F.G. Garvan and M.E.H. lsmail (eds.). Symbolic Computation. Number Theory. Special Functions. Physics and Combinatorics. 133-145. © 2001 Kluwer Academic Publishers.

134

SYMBOLIC COMPUTATION

and often write

[a, b, c, .. . j q]oo

:=

[aj q]oo[b; q]oo[c; q] oo . . .

It is easy to see that

(1) that

(2) and that

(3) Note that, as a function of z, [z; q]oo has an essential singularity at z = 0, no other singularities and simple zeros at z = qn for each n E Z. We say that a function

f : C\{O}

~

C

is q-elliptic if qE(i): f is analytic in C\{O} save for isolated poles and qE(ii): J(qz) = q-l J(z).

(qE(ii) is satisfied if, and only if, the differential I-form f(z)dz is invariant under z t-+ qz.) We say that points Zl, Z2 E C\{O} are q-equivalent if, for some nEZ, Z2 = qn Zl . The property of q-elliptic functions that we use is displayed in the simple

Lemma 1. If f : C\{O} ~ C is q-elliptic and P is a complete set of qinequivalent poles of f, then

I: res(J; 7r) = O.

'!rEP

Proof. See [7, §4].

In [4] the authors define three functions

a(q):=

L 00

ntm=-ex>

qn2+nm+m2,

(4)

The Borweins' Cubic Theta Functions

b(q):=

L 00

135

wn-mqn2+nm+m2

n,m=-oo

and

c(q):=

L 00

q(n+t)2+(n+t)(m+t)+(m+t)2.

n,m=-oo

and establish a number of relations between them. In particular, they found the following cubic analogue of Jacobi's quartic identity:

(5) In section 2 we will apply Lemma 1 to a certain q-elliptic function and thereby establish (Theorem 2) an old, very general, identity originally stated by Weierstrass. As an application of Theorem 2 we give a 2-parameter identity, closely related to that found by Winquist and we use this identity to prove some of the identities of [4] . In section 3 we apply Lemma 1 to a (different) q-elliptic function, thereby arriving at another proof of the identity of [2, (1)]. We use this identity to give expressions for a(q), b(q) and c(q) as sums of Lambert series.

2.

AN OLD IDENTITY It is shown in [5] how the following identity follows directly from Lemma

1:



ai

and aj are q- inequivalent for i i- j,

Then

(6) Several applications of Theorem 2 are given in [5]; in particular, the quintuple product identity, Winquist's identity and Jacobi's quartic identity are shown to follow immediately from this Theorem. Here, we use Theorem 2 to establish an identity, similar to Winquist's identity, which we use to give alternative proofs of the identities of [4].

136

SYMBOLIC COMPUTATION

Theorem 3. For any non-zero complex numbers x, y, (qj q)~[Xj q]~[y3j q3]00 _ (qj q)~[Yj q]~[X3j q3]00

= 3y(q3 j q3)~[Xj q]oo[Y; q]oo[xy; q]oo[Xy-lj q]oo = -3X(q3; q3)~[X; q]oo[Yj q]oo[xy; q]00[X- 1Yj q]oo. (7) Proof Take n

= 5 in Theorem 2 and

The hypotheses in Theorem 2 hold and with the help of (1), (2) and (3) we find that the various summands in (6) are: 1 -q -1 ,x -1 , x,.q]