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

Edited by Leun Kim

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].
 

Corollary. If $|q|<1$ then,
\begin{align*}
&\text{(i) }f(q,q) = 1+2\sum_{k=1}^\infty q^{k^2} = \frac{(-q; q^2)_\infty (q^2; q^2)_\infty}{(q; q^2)_\infty (-q^2; q^2)_\infty},\\
&\text{(ii) } f(q, q^3) = \sum_{k=0}^\infty q^{k(k+1)/2} = \frac{(q^2; q^2)_\infty}{(q; q^2)_\infty},\\
&\text{(iii) } f(-q, -q^2) = \sum_{k=0}^\infty (-1)^k q^{k(3k-1)/2} + \sum_{k=1}^\infty (-1)^k q^{k(3k+1)/2} = (q;q)_\infty.
\end{align*}
 

 
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.

 

 
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.



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