Ramanujan's cubic transformation and generalized modular equation

Abstract

We study the quotient of hypergeometric functions \begin{equation*} μ_{a}^*(r)=\fracπ{2\sin{(πa)}}\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 ${\mu }_{1/3}^{\ast }(r)$ by use of the properties of ${\mu }_a^{\ast }(r)$ and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.

16 pages