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Jacobi theta function

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For $|q|<1$ and $z\ne0$, the Jacobi triple product is
$$
\sum_{n\in\mathbb Z}z^nq^{n^2}
=\prod_{m\geq1}(1-q^{2m})(1+zq^{2m-1})(1+z^{-1}q^{2m-1}).
$$

Jacobi triple product

Nonvanishing of the Jacobi theta function

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / d / Solution

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The Jacobi triple product is an important identity in the theory of partitions and combinatorial mathematics. It relates the series expansion of certain infinite products and has applications in number theory, combinatorics, and the study of special functions.


Jacobi triple product

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Jacobi triple product

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