Relation between Ramanujan Theta Function and Jacobi Theta Function

Question

In the theory of $q-$series, we have Ramanujan Theta function \begin{align}\label{rama-theta} f(a,b):=\sum_{n=-\infty}^{\infty} a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}} ,\qquad |ab|<1. \end{align} And we also have Jacobi Theta function in complex analysis defined by \begin{align} \Theta(z|\tau)= \sum_{n=-\infty}^{\infty}e^{\pi i n^2 \tau}e^{2\pi i n z}. \end{align} We can also write it as \begin{align} \Theta(z|\tau)= \sum_{n=-\infty}^{\infty}q^{n^2}\eta^n, \end{align} where $\eta=e^{2\pi i z}$ and $q=e^{2\pi i \tau}.$

In wiki, https://en.wikipedia.org/wiki/Ramanujan_theta_function, he says "…… the Ramanujan theta function generalizes the form of the Jacobi theta functions……".

I cannot understand why it can be regarded as a generalization. In my opinion， Jacobi triple product have the following two expression \begin{align} \Theta(z|\tau)=\prod_{m=1}^{\infty}(1-e^{2\pi m i\tau})\left[1+e^{(2m-1)\pi i \tau+2\pi i z}\right]\left[1+e^{(2m-1)\pi i \tau-2\pi i z}\right]. \end{align} But in the notation of Ramanujan, we have \begin{align} f(a,b)=(-a;ab)_\infty(-b;ab)(ab;ab)_\infty. \end{align} This is more beautiful.

However, I still don't know what's the relation between the two kinds of "Theta function". Anybody can help me?

Answer 1

It is easy to see that the theta functions of Ramanujan and Jacobi are equivalent if we have $$ab=e^{2\pi i\tau}, a/b=e^{4\pi i z} $$ or $$a=e^{\pi i\tau+2\pi i z}, b=e^{\pi i\tau-2\pi i z} $$ Thus given any complex numbers $a, b$ with $|ab|<1$ gives us values of $\tau, z$ such that $\tau$ has positive imaginary part. The equations given above on the other hand express $a, b$ in terms of $\tau, z$.