Ramanujan's cubic transformation and generalized modular equation

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 $μ_{1/3}^*(r)$ by use of the properties of $μ_{a}^*(r)$ and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.

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