The identity in my recent answer can be stated in a particularly neat form: $${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\left({\frac{1}{2}, -n, n+1\atop{}};x^2\right).$$ Is this by any chance a partial case of something more general and/or has a straightforward proof?