The physics paper I am reading very non-chalantly defines the theta function as

$$ \theta(x;q) = (x;q)(q/x;q) \hspace{0.5in} \tilde{\theta}(x;q) = x^{-1/2}(x;q)(q/x;q) $$

where they are using the $q$-Pochammer symbol. This notation is a bit compressed it reads: $$ \theta(x;q) = \prod_{i=0}^\infty( 1 - xq^i) \prod_{i=0}^\infty( 1 - (q/x)q^i)$$

There are different versions of the Jacobi Triple Product Identity floating around. For example:

$$ \prod_{n > 0} (1 + q^{n-\frac{1}{2}}z)(1 + q^{n-\frac{1}{2}}z^{-1}) = \left( \sum_{l \in \mathbb{Z}} q^{\frac{l^2}{2} }z^l \right) \prod_{n > 0} \frac{1}{1-q^n}$$

However this doesn't seem to match up with the combination of q-Pochhammer symbols listed above.

How to expand $\theta(x;q)$ as an infinite series and recognize the infinite product? Wolfram Alpha was inconclusive.

  • 1
    $\begingroup$ Setting $x=-q^{1/2}z$ gets you further, but the Euler factor $(q; q)$ is then still missing; and the original theta functions tend to use $q$ (and $z$) squared. If only ratios of theta functions with the same $q$ are of interest, this shortcut works, but it overloads the notion of theta function even further. By the way, here are some helpful formulae. $\endgroup$ – ccorn Jun 4 '15 at 21:07

Ever since Jacobi introduced theta functions there have been a number of notations, variations and generalizations of them. Even Jacobi used more than one notation for theta functions. Read the Wikipedia article Jacobi theta functions (notational variations). The variations include choices of the two variables and constant terms or factors. You would only be confused to learn about all that.

A fairly standard form of one theta function is: $\theta_4(u,q):=\sum_{n=-\infty}^\infty (-1)^n \cos(2nu)\;q^{n^2},$ but also $\theta_4(u,q^{1/2})=(q;q)(x;q)(q/x;q)\;$ where $\;x=q^{1/2}e^{2iu}.$ Thus $\;\theta_4(u,q^{1/2})/(q;q)=(x;q)(q/x;q).\;$ I am not sure where the $x^{-1/2}$ comes from in $\tilde{\theta}(x;q)$.

  • $\begingroup$ i'm trying to take this variety of theta functions as a feature rather than a defect... however what happens when I am reading a paper and they choose one theta function over the other? $\endgroup$ – cactus314 Jan 6 '18 at 14:27
  • $\begingroup$ @cactus314 Please read the last paragraph of the Wikipedia article.. $\endgroup$ – Somos Jan 6 '18 at 15:08

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