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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[…] Product [2]. As before, we always denote as the Ramanujan Theta Function, which is defined in [3]. […]