A Unified Transformation Formula for Ramanujan's Theta Function
Mahipal Gurram
TL;DR
This paper addresses unifying Ramanujan's theta-function transformations by deriving a single closed formula for $f(\zeta a, \zeta b)$ with $\zeta$ a primitive $m$-th root of unity. Using residue-class dissections and modular substitutions, it establishes $f(\zeta a, \zeta b) = \sum_{k=0}^{m-1} \zeta^{k^2} a^{k(k+1)/2} b^{k(k-1)/2} f(A_m (ab)^{mk}, B_m (ab)^{-mk})$ with $A_m = a^{m(m+1)/2} b^{m(m-1)/2}$ and $B_m = a^{m(m-1)/2} b^{m+1)/2}$. The framework recovers Ramanujan's classical results for $m=2,3,4$—including even/odd dissections, the cubic transformation, and the compact quartic form—highlighting the underlying modular structure of $f(a,b)$. The unified formula provides a versatile tool for generating identities and exploring modular-type transformations of theta functions.
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 $f(ζa, ζb)$, where $ζ$ 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.