Entry 158

In the previous entry, given the Jacobi theta functions \(\vartheta_n(0,q)\) with nome \(q = e^{\pi i\tau}\), modular lambda function \(\lambda(\tau)\), and complete elliptic integral of the first kind \(K(k) = \tfrac{\pi}2\, {_2F_1}\big(\tfrac12,\tfrac12,1,k^2) \) we proposed three identities, one as,

$$\left(\frac{\vartheta_2(0,q)}{\sqrt{_2F_1\big(\tfrac12,\tfrac12,1,\lambda\big)}}\right)^4 \overset{\color{red}?}=\lambda$$ where \(\lambda = \lambda(\tau)\). Using the known definitions of \(\vartheta_n(0,q)\) and \(\lambda(\tau)\), the proposed identity implies 

$$\frac{_2F_1\big(\tfrac12,\tfrac12,1,\lambda\big)}{\eta^2(\tau)}=\left(\frac{\eta^2(\tau)}{\eta(\tfrac{\tau}2)\,\eta(2\tau)}\right)^4$$

For appropriate complex quadratics such as \(\tau=\sqrt{-n}\), then the RHS is a radical. But if the numerator of the LHS has a closed-form (implied in this list), then this also gives a closed-form for the denominator \(\eta^2(\tau)\) such as

$$\begin{align}\eta^2(\sqrt{-1}) &=\frac{\Gamma^2(\tfrac14)}{(2\pi)^{3/2}}\frac1{2^{1/2}}\\ \eta^2(\sqrt{-2}) &=\frac{\Gamma(\tfrac18)\Gamma(\tfrac38)}{(2\pi)^{3/2}}\frac1{2^{5/4}}\\ \eta^2(\sqrt{-3}) &=\frac{\Gamma^3(\tfrac13)}{(2\pi)^{2}}\frac{3^{1/4}}{2^{2/3}}\end{align}$$

and so on. Since an eta quotient is involved, then \(\eta(\sqrt{-d})\) for \(d\) with class number \(1\) will behave quite orderly and will be discussed in the next entry.