A Unified Transformation Formula for Ramanujan's Theta Function

TL;DR

This paper presents a unified transformation formula for Ramanujan's theta function, generalizing classical identities and providing systematic proofs for special cases involving roots of unity, modular substitutions, and dissections.

Contribution

It introduces a comprehensive transformation formula for the theta function that unifies and extends Ramanujan's classical identities using residue-class dissections and modular techniques.

Findings

Derived a closed-form transformation formula for $f( zeta a, zeta b)$

Recovered classical results for $m=2,3,4$ as special cases

Provided systematic proofs for Ramanujan's identities involving roots of unity

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 $f(\zeta a, \zeta b)$, where $\zeta$ 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.