Ramanujan’s cubic transformation and generalized modular equation

We study the quotient of hypergeometric functionsμ_a~*(r)=π/2sin(πa) F(a,1-a;1;1-r~3)/F(a,1-a;1;r~3),r∈(0,1)in the theory of Ramanujan's generalized modular equation for a ∈(0,1/2],and 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.