Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by: $$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$

I've found with Sage that for every $1\leq m \leq n \leq 80$ this polynomial has the property that all of its roots are real (negative, of course).

It seems these roots are not nice at all. For example for $m=3$ and $n=10$, one has $$P(t) = 120t^3 + 135 t^2 + 30t+1$$ and the roots are: $$ t_1 = -0.8387989...$$ $$ t_2 = -0.2457792...$$ $$ t_3 = -0.0404217...$$

Is it true that all roots of $P_{m,n}(t)$ are real?