Infinite products and normalized quotients of ypergeometric functions

For $r \in (0,1)$ and $a \in (0,1)$ the authors consider the quotient of hypergeometric functions $$ \mu_a(r)\equiv c F(a,1-a;1;1-r^2)/F(a,1-a;1;r^2), $$ where the normalizing coefficient $c = \pi/(2 \sin(\pi a)) .$ With this choice of $c,$ $\mu(r) \equiv \mu_{1/2}(r)$, where $\mu(r)$ is the modulus of the Grotzsch ring $B^2 \setminus [0,r]$ in the plane. A new infinite product expansion is given for $\mu(r).$ It is shown that several well-known properties of the function $\mu(r)$ have their counterparts for $\mu_a(r).$

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