Ramanujan's cubic transformation and generalized modular equation

TL;DR

This paper explores Ramanujan's cubic transformation and generalized modular equations, deriving an infinite product formula for a specific hypergeometric quotient and introducing a new cubic transformation formula.

Contribution

It presents a novel infinite product formula for =1/3 hypergeometric quotient and a new cubic transformation, expanding Ramanujan's classical results.

Findings

Derived an infinite product formula for =1/3 hypergeometric quotient

Established a new cubic transformation formula for hypergeometric functions

Enhanced understanding of Ramanujan's generalized modular equations

Abstract

We study the quotient of hypergeometric functions \begin{equation*} \mu_{a}^*(r)=\frac{\pi}{2\sin{(\pi a)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation for $a\in(0,1/2]$, find an infinite product formula for $\mu_{1/3}^*(r)$ by use of the properties of $\mu_{a}^*(r)$ and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.