TITLE: A hypergeometric identity related to Bessel functions
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: The coefficient of $x^n$ in $_2F_0(\alpha,\beta;z) {}_2F_0(\alpha,\beta;-z)$, multiplied by $n!$,
is 
$$\begin{aligned}
\sum_{k=0}^n (-1)^{n-k}\binom nk &(\alpha)_k(\beta)_k (\alpha)_{n-k}(\beta)_{n-k}\hfill\\
&=(-1)^n (\alpha)_n(\beta)_n\,
  {}_3F_2\left({-n,\atop}\!{\alpha,\atop 1-\alpha -n,}\,
  {\beta\atop1-\beta-n}\biggm | 1\right).
\end{aligned}
$$ 
The right side can be evaluated by Dixon's theorem to give the identity cited by Johannes Trost.