2 A first proof of (1.3) and a permanent modular equation

We begin with some well-known identities associated with $\vartheta _j(0|\tau )$ , $j=2,3,4$ .

Lemma 2.1. The following identities hold:

(2.1) $$ \begin{align} \!\!\!\!2\vartheta_2(0|2\tau)\vartheta_3(0|2\tau) &=\vartheta_2^2(0|\tau), \end{align} $$

(2.2) $$ \begin{align} \!\!\!\!\vartheta_3^2(0|2\tau) -\vartheta_3^2(0|\tau) &= - \vartheta_2^2(0|2\tau), \end{align} $$

(2.3) $$ \begin{align} \!\!\!\!\vartheta_3^2(0|2\tau) -\vartheta_2^2(0|\tau) &= \vartheta_4^2(0|2\tau), \end{align} $$

(2.4) $$ \begin{align} \!\!\!\!2\vartheta_3^2(0|2\tau) -\vartheta_3^2(0|\tau) &=\vartheta_4^2(0|\tau). \end{align} $$

Identities (2.1)–(2.4) play important roles in the study of the Gauss–Brent–Salamin algorithm and Ramanujan’s series for $1/\pi $ . (For more details, see [Reference Chan1].) Their proofs can be found in [Reference Chan1, page 58, (5.70)], [Reference Chan1, page 22, (2.26)], [Reference Chan1, page 22, (2.27)] and [Reference Chan1, Lemma 2.3], respectively.

Proof of (1.3).

By (1.2),

(2.5) $$ \begin{align}& 2\bigg(\sum_{m,n=-\infty}^\infty q^{2(m^2+mn +((N+1)/4)n^2)}\bigg)^2 \nonumber \\ &\qquad= 2[\vartheta_3(0|2\tau)\vartheta_3(0|2N\tau)+\vartheta_2(0|2\tau)\vartheta_2(0|2N\tau)]^2 \nonumber \\ &\qquad = 2\vartheta_3^2(0|2\tau)\vartheta_3^2(0|2N\tau) +2\vartheta_2^2(0|2\tau)\vartheta_2^2(0|2N\tau) \nonumber \\ &\qquad\qquad + 4 \vartheta_3(0|2\tau)\vartheta_3(0|2N\tau) \vartheta_2(0|2\tau)\vartheta_2(0|2N\tau) \nonumber \\ &\qquad = 2\vartheta_3^2(0|2\tau)\vartheta_3^2(0|2N\tau) +2\vartheta_2^2(0|2\tau)\vartheta_2^2(0|2N\tau) + \vartheta_2^2(0|\tau)\vartheta_2^2(0|N\tau), \end{align} $$

after two applications of (2.1). Applying (2.2) twice, the sum of the first two products in (2.5) is given by

(2.6) $$ \begin{align} \nonumber &2\vartheta_3^2(0|2\tau)\vartheta_3^2(0|2N\tau) +2\vartheta_2^2(0|2\tau)\vartheta_2^2(0|2N\tau)\\ \nonumber &\qquad=2\vartheta_3^2(0|2\tau)\vartheta_3^2(0|2N\tau)+2[\vartheta_3^2(0|2\tau)-\vartheta_3^2(0|\tau)] [\vartheta_3^2(0|2N\tau)-\vartheta_3^2(0|N\tau)]\\ \nonumber &\qquad=[2\vartheta_3^2(0|2\tau)-\vartheta_3^2(0|\tau)][2\vartheta_3^2(0|2N\tau)-\vartheta_3^2(0|N\tau)] + \vartheta_3^2(0|\tau)\vartheta_3^2(0|N\tau) \\ &\qquad=\vartheta^2_4(0|\tau)\vartheta^2_4(0|N\tau)+ \vartheta_3^2(0|\tau)\vartheta_3^2(0|N\tau), \end{align} $$

after applying (2.4) twice. Substituting (2.6) back into (2.5), we obtain (1.3) and this completes the proof.

Let

$$ \begin{align*}\lambda(\tau) = \dfrac{\vartheta_2^4(0|\tau)}{\vartheta_3^4(0|\tau)} \quad\text{and}\quad \beta_s=\lambda(s\tau).\end{align*} $$

A modular equation of degree s associated with $\lambda (\tau )$ is an algebraic relation satisfied by $\beta _1$ and $\beta _s$ . For example, the identity [Reference Chan1, Theorem 2.6]

(2.7) $$ \begin{align}\beta_2=\frac{\bigg(1-\sqrt{1-\beta_1}\bigg)^2}{\bigg(1+\sqrt{1-\beta_1}\bigg)^2} \end{align} $$

is a modular equation of degree 2. A modular equation of composite degrees 1,m,n,mn associated with $\lambda (\tau )$ is an algebraic relation involving $\beta _1, \beta _m,\beta _n$ and $\beta _{mn}.$ An example of a modular equation of composite degrees $1, 3, 13, 39$ , given by Ramanujan in his second letter to Hardy [Reference Ramanujan, Hardy, Aiyar and Wilson7, page xxix, (21)], is

(2.8) $$ \begin{align} &\dfrac{\{(1-\beta_1)(1-\beta_{39})\}^{1/8}-(\beta_1\beta_{39})^{1/8}}{\{(1-\beta_{3})(1-\beta_{13})\}^{1/8}-(\beta_3\beta_{13})^{1/8}} =\dfrac{1+\{(1-\beta_1)(1-\beta_{39})\}^{1/4}+(\beta_1\beta_{39})^{1/4}}{1+\{(1-\beta_{3})(1-\beta_{13})\}^{1/4}+(\beta_3\beta_{13})^{1/4}}.\\[-12pt]\nonumber \end{align} $$

A modular equation of degree s, or of composite degrees $1,m,n,mn$ , is called a permanent modular equation if the same relation is satisfied by more than one value of s, or by more than one pair of values $(m,n)$ , respectively. For example, in the same letter mentioned above, Ramanujan asserted that (2.8) is also true for $(m,n) = (5,11)$ and $(7, 9).$ We end this section by indicating that (2.6) can be interpreted as a permanent modular equation of composite degrees $1,2, \ell , 2\ell $ by using [Reference Chan1, (8.53)]

$$ \begin{align*}\dfrac{\vartheta_3^2(0|\tau)}{\vartheta_3^2(0|2\tau)} = \dfrac{2}{1+\sqrt{1-\beta_1}}.\end{align*} $$

The permanent modular equation is given by

(2.9) $$ \begin{align} (1+(\beta_2\beta_{2\ell})^{1/4})^2 =2\dfrac{1+\sqrt{\beta_1\beta_\ell}+\sqrt{(1-\beta_1)(1-\beta_\ell)}}{(1+\sqrt{1-\beta_1})(1+\sqrt{1-\beta_\ell})}.\end{align} $$

Identity (2.9) can be transformed to a relation involving only $\beta _1$ and $\beta _\ell $ by using the modular equation (2.7) of degree 2 to eliminate $\beta _2$ and $\beta _{2\ell }$ . This results in a ‘trivial’ relation which is essentially the identity

$$ \begin{align*}\bigg(\sqrt{(1+x)(1+y)}+\sqrt{(1-x)(1-y)}\bigg)^2 =2\bigg(1+\sqrt{(1-x^2)(1-y^2)}+xy\bigg).\end{align*} $$

3 Second proof of (1.3)

Our main theorem in this section is the following result.

Theorem 3.1. Let

(3.1) $$ \begin{align} {A}_{\alpha , \beta}(u,v|\tau)&=\sum_{m,n=-\infty}^\infty q^{\alpha m^2+\alpha mn+\beta n^2}e^{2iu(2m+n)} e^{2ivn}, \end{align} $$

where $\alpha $ and $\beta $ are integers satisfying $0<\alpha <4\beta $ , so that the double sum converges. Then, for any integer M and even integer N satisfying $0< M < N$ ,

(3.2) $$ \begin{align} \nonumber &2{A}_{2M , N/2}(u+v, w+x|\tau){A}_{2M , N/2}(u-v, w-x|\tau)\\ &\qquad =\sum_{i=1}^{4}\vartheta_i(2u|M\tau)\vartheta_i(2v|M\tau) \vartheta_i(2w|(N-M)\tau)\vartheta_i(2x|(N-M)\tau). \end{align} $$

In addition to establishing (1.3), Theorem 3.1 also leads to other analogues of (1.3). We give a sample of such identities as follows:

$$ \begin{align*}& 2\bigg(\sum_{m,n=-\infty}^\infty (-1)^n q^{2(m^2+mn+((N+1)/4)n^2)}\bigg)^2 \\ &\qquad =-\vartheta_2^2(0|\tau)\vartheta_2^2(0|N\tau)+\vartheta_3^2(0|\tau)\vartheta_3^2(0|N\tau)+\vartheta_4^2(0|\tau)\vartheta_4^2(0|N\tau),\\& 2\bigg(\sum_{m,n=-\infty}^\infty q^{2((m+1/2)^2+(m+1/2)n+((N+1)/4)n^2)}\bigg)^2 \\ &\qquad =\vartheta_2^2(0|\tau)\vartheta_2^2(0|N\tau)+\vartheta_3^2(0|\tau)\vartheta_3^2(0|N\tau)-\vartheta_4^2(0|\tau)\vartheta_4^2(0|N\tau), \\& 2\bigg(\sum_{m,n=-\infty}^\infty (-1)^n q^{2((m+1/2)^2+(m+1/2)n+((N+1)/4)n^2)}\bigg)^2 \\ &\qquad =-\vartheta_2^2(0|\tau)\vartheta_2^2(0|N\tau)+\vartheta_3^2(0|\tau)\vartheta_3^2(0|N\tau)-\vartheta_4^2(0|\tau)\vartheta_4^2(0|N\tau). \end{align*} $$

We next prove Theorem 3.1. To establish Theorem 3.1, we need several identities involving Jacobi’s theta functions defined in Section 1. These identities are extensions of identities in Lemma 2.1 and they can all be found in [Reference Lawden6, Ch. 1].

Lemma 3.3. The following identities hold:

(3.3) $$ \begin{align} &\vartheta_3(u+v|2\tau)\vartheta_2(u-v|2\tau) -\vartheta_2(u+v|2\tau)\vartheta_3(u-v|2\tau) =\vartheta_1(u|\tau)\vartheta_1(v|\tau), \end{align} $$

(3.4) $$ \begin{align} &\vartheta_3(u+v|2\tau)\vartheta_2(u-v|2\tau) +\vartheta_2(u+v|2\tau)\vartheta_3(u-v|2\tau) =\vartheta_2(u|\tau)\vartheta_2(v|\tau), \end{align} $$

(3.5) $$ \begin{align} &\vartheta_3(u+v|2\tau)\vartheta_3(u-v|2\tau) -\vartheta_3(u|\tau)\vartheta_3(v|\tau) = - \vartheta_2(u+v|2\tau)\vartheta_2(u-v|2\tau), \end{align} $$

(3.6) $$ \begin{align} &\vartheta_3(u+v|2\tau)\vartheta_3(u-v|2\tau) -\vartheta_2(u+v|\tau)\vartheta_2(u-v|\tau) = \vartheta_4(u|2\tau)\vartheta_4(v|2\tau), \end{align} $$

(3.7) $$ \begin{align} &2\vartheta_3(u+v|2\tau)\vartheta_3(u-v|2\tau) -\vartheta_3(u|\tau)\vartheta_3(v|\tau) =\vartheta_4(u|\tau)\vartheta_4(v|\tau).\qquad \end{align} $$

We are now ready to prove Theorem 3.1.

Proof of Theorem 3.1.

From [Reference Chan, Chan, Liu and Zudilin2, (5.7)] and the proof of [Reference Chan, Chan, Liu and Zudilin2, Theorem 5.2],

(3.8) $$ \begin{align} {A}_{\alpha,\beta}(u,v|\tau) = \vartheta_3(2u|\alpha\tau)\vartheta_3(2v|(4\beta-\alpha)\tau) +\vartheta_2(2u|\alpha\tau)\vartheta_2(2v|(4\beta-\alpha)\tau). \end{align} $$

Therefore,

(3.9) $$ \begin{align} \nonumber &2{A}_{2M , N/2}(u+v, w+x|\tau){A}_{2M , N/2}(u-v, w-x|\tau)\\ \nonumber &\qquad = 2[\vartheta_3(2u+2v|2M\tau)\vartheta_3(2w+2x|(2N-2M)\tau) \\ \nonumber &\qquad\qquad +\vartheta_2(2u+2v|2M\tau)\vartheta_2(2w+2x|(2N-2M)\tau)]\\ \nonumber &\qquad \quad \times [\vartheta_3(2u-2v|2M\tau)\vartheta_3(2w-2x|(2N-2M)\tau) \\ \nonumber &\qquad\qquad +\vartheta_2(2u-2v|2M\tau)\vartheta_2(2w-2x|(2N-2M)\tau)]\\ \nonumber &\qquad =:2[AB+CD][EF+GH]\\ &\qquad =2ABEF+2CDGH + 2ABGH+2CDEF, \end{align} $$

where

$$ \begin{align*} A=\vartheta_3(2(u+v)|2M\tau),\quad &B=\vartheta_3(2(w+x)|2(N-M)\tau), \\ C=\vartheta_2(2(u+v)|2M\tau),\quad &D=\vartheta_2(2(w+x)|2(N-M)\tau),\\ E=\vartheta_3(2(u-v)|2M\tau),\quad &F=\vartheta_3(2(w-x)|2(N-M)\tau),\\ G=\vartheta_2(2(u-v)|2M\tau),\quad &H=\vartheta_2(2(w-x)|2(N-M)\tau). \end{align*} $$

Further, let

$$ \begin{align*} J=\vartheta_3(2u|M\tau)\vartheta_3(2v|M\tau),&\quad K=\vartheta_3(2v|M\tau), \\ L=\vartheta_3(2w|(N-M)\tau),&\quad P=\vartheta_3(2x|(N-M)\tau). \end{align*} $$

Then, by two applications of (3.5), the sum of the first two products in (3.9) becomes

(3.10) $$ \begin{align} \nonumber &2ABEF+2CDGH =2ABEF+2[AE-JK][BF-LP]\\ \nonumber &\qquad =[2AE-JK][2BF-LP] + JKLP \\ \nonumber &\qquad =\vartheta_4(2u|M\tau)\vartheta_4(2v|M\tau)\vartheta_4(2w|(N-M)\tau)\vartheta_4(2x|(N-M)\tau)\\ &\qquad\qquad + \vartheta_3(2u|M\tau)\vartheta_3(2v|M\tau)\vartheta_3(2w|(N-M)\tau)\vartheta_3(2x|(N-M)\tau), \end{align} $$

after applying (3.7) twice.

For the sum of the next two products in (3.9), we find that

(3.11) $$ \begin{align} \nonumber &2ABGH+2CDEF \\ \nonumber & \qquad = [AG-CE][BH-DF]+[AG+CE][BH+DF] \\ \nonumber &\qquad =\vartheta_1(2u|M\tau)\vartheta_1(2v|M\tau)\vartheta_1(2w|(N-M)\tau)\vartheta_1(2x|(N-M)\tau) \\ &\qquad\qquad + \vartheta_2(2u|M\tau)\vartheta_2(2v|M\tau)\vartheta_2(2w|(N-M)\tau)\vartheta_2(2x|(N-M)\tau), \end{align} $$

after two applications each of (3.3) and (3.4). Substituting (3.10) and (3.11) back into (3.9), we obtain (3.2) and this completes the proof of Theorem 3.1.

4 A generalisation of an identity of Chan, Chua and Solé

In [Reference Chan, Chua and Solé5, Theorem 1.1], Chan, Chua and Solé discovered the identity

(4.1) $$ \begin{align} \bigg(\sum_{m,n=-\infty}^\infty q^{2(m^2+mn+2n^2)}\bigg)^2 &=\bigg(\sum_{m,n=-\infty}^\infty (-1)^{m+n}q^{2(m^2+mn+2n^2)}\bigg)^2 \notag\\ &\qquad+\bigg(\sum_{m,n=-\infty}^\infty q^{2((m+1/2)^2+(m+1/2)n+2n^2)}\bigg)^2 \end{align} $$

and gave a proof by considering 7-modular lattices from codes over $\mathbf F_2\times \mathbf F_2.$ Shortly after the discovery of (4.1), they [Reference Chan, Chua and Solé4, Lemma 2.2(ii)] found an infinite family of identities, parametrised by d, extending (4.1), given by

(4.2) $$ \begin{align} \bigg(\sum_{m,n=-\infty}^\infty q^{2(m^2+mn+dn^2)}\bigg)^2 &=\bigg(\sum_{m,n=-\infty}^\infty (-1)^{m+n}q^{2(m^2+mn+dn^2)}\bigg)^2 \notag\\ &\qquad+\bigg(\sum_{m,n=-\infty}^\infty q^{2((m+1/2)^2+(m+1/2)n+dn^2)}\bigg)^2. \end{align} $$

Recently, motivated by the study of 15-modular lattices, Chan et al. [Reference Chan, Chan and Solé3] discovered the identity

(4.3) $$ \begin{align} \bigg(\sum_{m,n=-\infty}^\infty q^{2m^2+mn+2n^2}\bigg)^2 &=\bigg(\sum_{m,n=-\infty}^\infty (-1)^{m+n}q^{2m^2+mn+2n^2}\bigg)^2 \notag\\ &\qquad+\bigg(2\sum_{m,n=-\infty}^\infty q^{2(2(m+1/2)^2+(m+1/2)n+2n^2)}\bigg)^2. \end{align} $$

They proved (4.3) by establishing an infinite family of identities [Reference Chan, Chan and Solé3, Theorem 1.1], parametrised by b and d, extending (4.3), given by

(4.4) $$ \begin{align} \bigg(\sum_{m,n=-\infty}^\infty q^{dm^2+bmn+dn^2}\bigg)^2 &=\bigg(\sum_{m,n=-\infty}^\infty (-1)^{m+n}q^{dm^2+bmn+dn^2}\bigg)^2 \notag\\ &\qquad+\bigg(2\sum_{m,n=-\infty}^\infty q^{2(d(m+1/2)^2+b(m+1/2)n+dn^2)}\bigg)^2, \end{align} $$

where $1\leq b\leq d-1$ . Motivated by (4.4), they further generalised (4.2) by introducing another family of identities depending on the parameters b and d [Reference Chan, Chan and Solé3, (1.6)], given by

(4.5) $$ \begin{align} \bigg(\sum_{m,n=-\infty}^\infty q^{2(bm^2+bmn+dn^2)}\bigg)^2 &=\bigg(\sum_{m,n=-\infty}^\infty (-1)^{m+n}q^{2(bm^2+bmn+dn^2)}\bigg)^2 \notag\\ &\qquad+\bigg(\sum_{m,n=-\infty}^\infty q^{2(b(m+1/2)^2+b(m+1/2)n+dn^2)}\bigg)^2. \end{align} $$

Both (4.4) and (4.5) turn out to admit two variable extensions [Reference Chan, Chan, Liu and Zudilin2, Theorems 5.1 and 5.2]. For example, the two variable extension of (4.5) is given by

(4.6) $$ \begin{align} A_{\alpha,\beta}^2(u,v|\tau)-C_{\alpha,\beta}^2(u,b|\tau) = A_{\alpha,\beta}^2(u,v+\pi/2|\tau)-C_{\alpha,\beta}^2(u,v+\pi|\tau), \end{align} $$

where $A_{\alpha ,\beta }(u,v|\tau )$ is defined in (3.1) and

(4.7) $$ \begin{align} {C}_{\alpha , \beta}(u,v|\tau) =\sum_{m,n=-\infty}^\infty q^{\alpha m^2+ \alpha mn+\beta n^2+\alpha m+\alpha n/2+\alpha /4} e^{2iu(2m+n+1)} e^{2ivn}. \end{align} $$

The discovery of (4.6) was motivated by the generalisation of Borweins’ cubic theta function identity discovered by Schultz [Reference Schultz8, Corollary 4.3], [Reference Chan, Chan, Liu and Zudilin2, (1.4)]. We now present a two variable extension of (4.6).

Theorem 4.1. Let ${A}_{\alpha ,\beta }(u,v|\tau )$ and ${C}_{\alpha , \beta }(u,v|\tau )$ be defined in (3.1) and (4.7), respectively. Suppose $0<\alpha <4\beta $ , so that the double series converge. Then,

(4.8) $$ \begin{align} &{A}_{\alpha,\beta}(u,w-u|\tau){A}_{\alpha,\beta}(v,x-v|\tau) +{C}_{\alpha,\beta}(v,w-u+\pi/2|\tau){C}_{\alpha,\beta}(u,x-v+\pi/2|\tau)\nonumber\\ &\qquad= {A}_{\alpha,\beta}(u,w-u+\pi/2|\tau){A}_{\alpha,\beta}(v,x-v+\pi/2|\tau) +{C}_{\alpha,\beta}(v,w-u|\tau){C}_{\alpha,\beta}(u,x-v|\tau). \end{align} $$

Proof. Since the proof is a straightforward computation, we omit the details and present only the key steps.

Identity (3.8) allows us to express $A_{\alpha ,\beta }(u,v|\tau )$ in terms of $\vartheta _2(\cdot |\tau )$ and $\vartheta _3(\cdot |\tau )$ . Together with the easy facts that

(4.9) $$ \begin{align} \vartheta_2(u+\pi|\tau) = -\vartheta_2(u|\tau) \quad \text{and} \quad \vartheta_3(u+\pi|\tau) = \vartheta_3(u|\tau), \end{align} $$

one can show that

(4.10) $$ \begin{align} \nonumber &\hspace{-6pt}{A}_{\alpha,\beta}(u,w-u|\tau){A}_{\alpha,\beta}(v,x-v|\tau) -{A}_{\alpha,\beta}(u,w-u+\pi/2|\tau){A}_{\alpha,\beta}(v,x-v+\pi/2|\tau) \\ \nonumber &\hspace{-6pt}\qquad=2\vartheta_3(2u|\alpha\tau)\vartheta_3(2(w-u)|(4\beta-\alpha)\tau) \vartheta_2(2v|\alpha\tau)\vartheta_2(2(x-v)|(4\beta-\alpha)\tau) \\ &\hspace{-6pt}\qquad\qquad + 2\vartheta_2(2u|\alpha\tau)\vartheta_2(2(w-u)|(4\beta-\alpha)\tau) \vartheta_3(2v|\alpha\tau)\vartheta_3(2(x-v)|(4\beta-\alpha)\tau). \end{align} $$

Next, from [Reference Chan, Chan, Liu and Zudilin2, (5.7)] and the proof of [Reference Chan, Chan, Liu and Zudilin2, Theorem 5.2], we see that

$$ \begin{align*} {C}_{\alpha,\beta}(u,v|\tau) = \vartheta_2(2u|\alpha\tau)\vartheta_3(2v|(4\beta-\alpha)\tau) +\vartheta_3(2u|\alpha\tau)\vartheta_2(2v|(4\beta-\alpha)\tau). \end{align*} $$

This, together with (4.9), then enables one to show that

$$ \begin{align*} \nonumber {C}_{\alpha,\beta}(v,w-u|\tau) {C}_{\alpha,\beta}(u,x-v|\tau) - {C}_{\alpha,\beta}(v,w-u+\pi/2|\tau) {C}_{\alpha,\beta}(u,x-v+\pi/2|\tau) \end{align*} $$

takes the same form as the right-hand side of (4.10). Thus, we obtain (4.8) and this completes the proof of the theorem.