Ramanujan’s cubic transformation and generalized modular equation

We study the quotient of hypergeometric functions $\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 )}},r \in (0,1) $ in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2], find an infinite product formula for µ1/3*(r) by use of the properties of µ a * and Ramanujan’s cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan’s cubic transformation.