A Unified Transformation Formula for Ramanujan's Theta Function
Mahipal Gurram

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 , 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 , where is a primitive root of unity . As special cases, we recover and systematically prove Ramanujan's classical results for and , including even odd dissections, cubic transformation and the compact quartic form involving complex coefficients.
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Taxonomy
TopicsAdvanced Mathematical Identities · Advanced Combinatorial Mathematics · Analytic Number Theory Research
