Treffer: Theta-function identities and the explicit formulas for theta-function and their applications

Let \(h_{k,n}= \frac{\varphi(e^{-\pi\sqrt{n/k}})} {k^{1/4}\varphi(e^{-\pi\sqrt{nk}})}\) and \(h_{k,n}'= \frac{\varphi(-e^{-2\pi\sqrt{n/k}})} {k^{1/4}\varphi(-e^{-2\pi\sqrt{nk}})},\) where \(\varphi(q)=\sum_{j=-\infty}^\infty q^{j^2}\). Properties of \(h_{k,n}\) and \(h_{k,n}'\) are studied, and \(h_{k,n}\) and \(h_{k,n}'\) are explicitly evaluated for several values of \(k\) and \(n\). These evaluations come from so-called P-Q modular equations, some of which were given by Ramanujan and others are established using results in his notebooks. The paper ends with explicit values for \(\varphi(e^{-k\pi})\) and \(\varphi(-e^{-2k\pi})\) for \(1\leq k \leq 6\).