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

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.

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

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