The Ramanujan theta function $$f(a,b) = \sum_{n \in \mathbb Z} a^{n(n+1)/2} b^{n(n-1)/2}$$ satisfies the following Jacobi triple product identity $f(a,b) = (-a;ab)\_\infty (-b;ab)\_\infty (ab;ab)\_\infty$.

Are there any three variable generalizations which also satisfy identities like that?

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I do not know what it is for, as any basic hypergeometric summation can be interpretted as a certain multi-variate "Jacobi-like" identity. Probably, the easiest 4-variate example, known in the theory of basic hypergeometric functions as Ramanujan's summation (see, for example, Section 5.2 in the $q$-Bible of Gasper and Rahman), is $$ \sum_{n\in\mathbb Z}\frac{(a;q) _ n}{(b;q) _ n}\biggl(\frac ca\biggr)^n =\frac{(q,b/a,c,q/c;q) _ \infty}{(b,q/a,c/a,b/c;q) _ \infty}, $$ where the standard notation $(a;q) _ n=\prod_{k=1}^n(1-aq^{k-1})$ is used (and extending this symbol to the negative $n$ as well). The formula is valid for $|q|<1$ and $|b|<|c|<|a|$. The Jacobi triple product identity can be realised as a limiting case of this summation.

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