Ninth degree analogue of Ramanujan's septic theta function identity

Abstract

On page 206 in his lost notebook, Ramanujan recorded a seventh degree identity for his theta function $\ddot{I}†(q)$. We give an analogous ninth degree identity. We also provide an application of an entry from his second notebook on a cubic equation and an interpretation with theta functions for some of his trigonometric identities. Lastly, we calculate five examples for $\ddot{I}†(e^{-\ddot{I}€\sqrt{n}})$.

21 pages, submitted