There is a beautiful cubic analogue of Jacobi’s fundamental theta function identity:
θ
3
4
=
θ
4
4
+
θ
2
4
\theta _3^4 = \theta _4^4 + \theta _2^4
. It is
\[
(
∑
n
,
m
=
−
∞
∞
q
n
2
+
n
m
+
m
2
)
3
=
(
∑
n
,
m
=
−
∞
∞
ω
n
−
m
q
n
2
+
n
m
+
m
2
)
3
+
(
∑
n
,
m
=
−
∞
∞
q
(
n
+
1
3
)
2
+
(
n
+
1
3
)
(
m
+
1
3
)
+
(
m
+
1
3
)
2
)
3
.
{\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m^2}}}} } \right )^3} = {\left ({\sum \limits _{n,m = - \infty }^\infty {{\omega ^{n - m}}{q^{{n^2} + nm + {m^2}}}} } \right )^3} + {\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{{(n + \frac {1}{3})}^2} + (n + \frac {1}{3})(m + \frac {1}{3}) + {{(m + \frac {1}{3})}^2}}}} } \right )^3}.
\]
Here
ω
=
exp
(
2
π
i
/
3
)
\omega = \exp (2\pi i/3)
. In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.