trigonometric → elliptic in the infinite product representation of θ3(z,q)

If G(q) = \prod_{n=1}^{\infty} (1-q^{2n}), then

\theta_{3}(z,q) = G(q) \prod_{n=1}^{\infty} \left[1 - 2q^{2n-1}\cos(2z) +q^{4n-2}\right]

Let’s define an analogue of the theta function \theta_{3}(z,q) by replacing that trigonometric cosine \cos with an elliptic \mathrm{cn}(2z,k), where k is the elliptic modulus.

So we’d have:

A_{3}(z,q;k) = G(q) \prod_{n=1}^{\infty} \left[1 - 2q^{2n-1}\mathrm{cn}(2z;k) +q^{4n-2}\right]

Here’s a domain colored picture of A_{3}(z,1/2;1/2):

Advertisements