[Calculation 15] Introduction to the Ramanujan Theta Functions

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.
 

 
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.