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arXiv · 2511.09565

A Unified Transformation Formula for Ramanujan's Theta Function

Mahipal Gurram
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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.

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Mahipal Gurram. 2025-11-10. A Unified Transformation Formula for Ramanujan's Theta Function. https://arxiv.org/abs/2511.09565

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