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?


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 $\tau, z$.


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