On the relationship between hypergeometric functions and some multivalued functions (Q2387958)

Let \(\vartheta_1(\tau)=\vartheta(\tau;1,1,3\sqrt{5})\) and \(\vartheta_2(\tau)=\vartheta(\tau;2,1,3\sqrt{5})\) be Hecke theta functions in the field \({\mathbb Q}(\sqrt{5})\) and let \(\eta(\tau)\) be Dedekind's eta function. The authors relate \(\frac{\vartheta_1(\tau)}{\eta^2(\tau)}\) and \(\frac{\vartheta_2(\tau)}{\eta^2(\tau)}\) to the fundamental matrix of the Gauss equation \[ z(z-1)w''+((1+a+b)z-c)w'+abw=0 \] in a neighborhood of \(z=0\) and in a neighborhood of \(z=1\).