Relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

Question

What are the relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

Specifically, I need a transformation that transforms:

$_2F_1\left(a,b;c; -\sinh^2(x)\right)$ to $_2F_1\left(a',b';c'; \cosh^{-2}(x)\right)$

Thanks in advance.

Answer 1

As pointed out in the comments, if we use DLMF 15.8.3 and $a\neq b$ then $$ F\left({a,b\atop c};-\sinh^2x\right)=\frac{(b)_{-a}}{(c)_{-a}}\cosh^{-2a}x\ F\left({a,c-b\atop a-b+1};\cosh^{-2}x\right) +\frac{(a)_{-b}}{(c)_{-b}}\cosh^{-2b}x\ F\left({b,c-a\atop b-a+1};\cosh^{-2}x\right). $$ If $a=b$ we end up with an indeterminate form that will require L'Hôpital.