Five years ago, there was a question about the positivity of the hypergeometric functions $_{1}F_{2}$ on a short interval $(0,5)$. I would like to ask a very similar question but for the hypergeometric functions $_{2}F_{1}$ on the whole line. The question is
whether or not the following $$_2F_1\Big[\frac{3}{2}, \frac{3}{2}; 2 ;- x^2 \Big] > 0 \quad \text{for all } x>0$$ holds.
Via Maple, plotted on the interval $[0,20]$, it seems that the answer is affirmative, see the picture below, as the function decays to zero.
I tried with Wolfram Alpha, please take a look.