[Calculation 15] Introduction to the Ramanujan Theta Functions

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Edited by Leun Kim

In this post, we introduce the Ramanujan theta functions $f(a,b)$, which generalize the form of the Jacobi theta functions. Here we define the Ramanujan theta function, and introduce some elementary properties.

First, we define the Ramanujan theta function as
$$
f(a,b) = 1+\sum_{k=1}^\infty (ab)^{k(k-1)/2} (a^k + b^k) = \sum_{k=-\infty}^\infty a^{k(k+1)/2} b^{k(k-1)/2}
$$
where $|ab|<1$. Then the following holds.
 

Theorem. We have
\begin{align*}
&\text{(i) } f(a,b) = f(b,a),\\
&\text{(ii) } f(1,a) = 2f(a,a^3),\\
&\text{(iii) } f(-1,a) = 0,\\
&\text{(iv) } \forall n \in \mathbb Z, \; f(a,b) = a^{n(n+1)/2} b^{n(n-1)/2} f(a(ab)^n, b(ab)^{-n}).
\end{align*}
 

 
Proof. (i) is trivial. For (ii), we have
\begin{eqnarray*}
f(1,a)
&=&
2+\sum_{k=1}^\infty a^{k(k+1)/2} + \sum_{k=2}^\infty a^{k(k-1)/2}\\
&=&
2 \left( 1+ \sum_{k=1}^\infty a^{k(k+1)/2}\right)\\
&=&
2 \left(1+ \sum_{k=1}^\infty a^{k(2k+1)}+\sum_{k=1}^\infty a^{k(2k-1)} \right)\\
&=&
2f(a,a^3).
\end{eqnarray*}
For (iii), we have
\begin{eqnarray*}
f(-1,a)
&=&
\sum_{k=2}^\infty (-1)^{k(k+1)/2} a^{k(k-1)/2}
+
\sum_{k=1}^\infty (-1)^{k(k-1)/2} a^{k(k+1)/2}\\
&=&
\sum_{k=1}^\infty (-1)^{(k+1)(k+2)/2} a^{k(k+1)/2}
+
\sum_{k=1}^\infty (-1)^{k(k-1)/2} a^{k(k+1)/2}\;\;=\;\;0.
\end{eqnarray*}
And for (iv), we have
\begin{eqnarray*}
f(a,b)
&=&
\sum_{k=-\infty}^\infty a^{k(k+1)/2} b^{k(k-1)/2}\\
&=&
\sum_{k=-\infty}^\infty a^{(k+n)(k+n+1)/2} b^{(k+n)(k+n-1)/2}\\
&=&
a^{n(n+1)/2} b^{n(n-1)/2} \sum_{k=-\infty}^\infty a^{k(k+2n+1)/2} b^{k(k+2n-1)/2}\\
&=&
a^{n(n+1)/2} b^{n(n-1)/2} \sum_{k=-\infty}^\infty (a(ab)^n)^{k(k+1)/2} (b(ab)^{-n})^{k(k-1)/2}\\
&=&
a^{n(n+1)/2} b^{n(n-1)/2} f(a(ab)^n, b(ab)^{-n}).
\end{eqnarray*}

 
References.
[1] Bruce C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, pp. 34-35.

 
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Leun Kim

Ph.D Candidate at The University of Tokyo
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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