[Calculation 18] Some Useful Formulas from the Jacobi Triple Product

In this post, we introduce some useful formulas, which can be proved by the Jacobi Triple Product [2]. As before, we always denote $f$ as the Ramanujan Theta Function, which is defined in [3].
 

 
Proof. We recall the Jacobi Triple Product Formula:
\begin{equation}\tag{1}
f(a,b) = (-a; ab)_\infty (-b; ab)_\infty (ab; ab)_\infty.
\end{equation}
(i) The first equality is trivial from the definition of the Ramanujan Theta Function. For the second equality, replacing $a=b=q$ in (1), we obtain
\begin{eqnarray*}
f(q,q)
&=&
(-q; q^2)_\infty^2 (q^2; q^2)_\infty \\
&=&
\frac{(q^2; q^2)_\infty}{(q; q^2)_\infty^2 (-q^2; q^2)_\infty^2}\\
&=&
\frac{1}{(q; q^2)_\infty (-q^2; q^2)_\infty} \frac{(q^2; q^2)_\infty}{(q; q^2)_\infty (-q^2; q^2)_\infty} \\
&=&
\frac{1}{(q; q^2)_\infty (-q^2; q^2)_\infty} \frac{(q; q)_\infty (-q; q)_\infty}{(q;q^2)_\infty (-q^2; q^2)_\infty}\\
&=&
\frac{1}{(q; q^2)_\infty (-q^2; q^2)_\infty} (q^2; q^2)_\infty (-q; q^2)_\infty.
\end{eqnarray*}
 
(ii) The first equality is trivial from the definition of the Ramanujan Theta Function. And from the equation (1), we have
\begin{eqnarray*}
f(q; q^3)
&=&
(-q; q^4)_\infty (-q^3; q^4)_\infty (q^4; q^4)_\infty\\
&=&
(-q; q^2)_\infty (-q^2; q^2)_\infty (q^2; q^2)_\infty\\
&=&
\frac{(q^2; q^2)_\infty}{(q; q^2)_\infty}
\end{eqnarray*}
where we used
$$
(-q; q^4)_\infty (-q^3; q^4)_\infty = (-q; q^2)_\infty
$$
for the first equality and
$$
(-q; q^2)_\infty = \frac{1}{(q; q^2)_\infty (-q^2; q^2)_\infty}
$$
for the second equality.
 
(iii) The first equality is also tirival. For the second one, the equation (1) yields
$$
f(-q, -q^2) = (q;q^3)_\infty (q^2; q^3)_\infty (q^3; q^3)_\infty = (q; q)_\infty.
$$
$\square$
 
References.
[1] Bruce C. Berndt, Ramanujan’s Notebooks Part III, Springer-Verlag, pp. 36-38.
[2] Leun Kim, [Calculation 17] Jacobi Triple Product Formula.
[3] Leun Kim, [Calculation 15] Introduction to the Ramanujan Theta Functions.