A Unified Transformation Formula for Ramanujan's Theta Function

Abstract

In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class dissections and modular substitutions, we obtain a closed transformation formula for $f(\hat{I}\P a,\hat{I}\P b)$, where $\hat{I}\P$ is a primitive root of unity $m$. As special cases, we recover and systematically prove Ramanujan's classical results for $m=2,3,$ and $4$, including even odd dissections, cubic transformation and the compact quartic form involving complex coefficients.