Ninth degree analogue of Ramanujan's septic theta function identity

TL;DR

This paper extends Ramanujan's work by deriving a ninth degree theta function identity, providing applications to cubic equations, and illustrating results with explicit examples.

Contribution

It introduces a new ninth degree identity for Ramanujan's theta function, expanding the mathematical understanding of these functions and their applications.

Findings

Derived a ninth degree theta function identity similar to Ramanujan's seventh degree identity.

Connected Ramanujan's identities to cubic equations and trigonometric interpretations.

Computed explicit values of $\

Abstract

On page 206 in his lost notebook, Ramanujan recorded a seventh degree identity for his theta function $\varphi(q)$. 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 $\varphi(e^{-\pi\sqrt{n}})$.