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This is a cross-post from MSE since there wasn't having enough attention. I need your help on proving the following identity Theorem Let $q=e^{-\pi \frac{K'}{K}}$ where $K$ denotes the complete ...

Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...