On the modular functions arising from the Theta constants
On the modular functions arising from the Theta constants
Some modular functions arising from the theta constants $ \vartheta_2(\tau)$, $ \vartheta_3(\tau)$, $ \vartheta_4(\tau)$ are investigated. Let $n$ be an odd square-free positive integer as in [4,7]. It is obtained a necessary and sufficient condition that $ \varphi_{\delta,\rho,3}(\tau)=\prod_{\delta|n,\rho|n}\Big({\vartheta_3(\delta\tau) \over\vartheta_3(\rho\tau)}\Big)^{r_\delta}$ is invariant with respect to transformations in $ \theta(n)$. Also, It …