Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.