In this entry I asked for the real-rootedness of a polynomial, and two very interesting answers were given: one using Malo's theorem and the other a clever rewriting of the expression using Jacobi Polynomials which are known to be real-rooted.

Now, in the middle of a reasoning I had to somehow prove that the following polynomials are real-rooted:

$$Q_{m,n}(t) = \sum_{j=0}^m \binom{m}{j}\binom{n}{j} \binom{t-j}{m+n+1}$$

defined for $m\leq n$, integers.

Observe that where in the last question one had $t^j$, here one has a $\binom{t-j}{m+n+1}$.

Of course, it has been verified numerically for the first small cases and it is true and, in fact, it seems to be true that all the roots are $\leq m+n$. It's possible to prove that many of the roots are integers, which suggests one may "factor out" something like $\binom{t-m}{n+1}$ out of the expression. The problem is that the remaining factor does not seem to be friendly at all.

This leads me to the following questions:

1) Any ideas to prove the real-rootedness of this thing?

2) (Much wider and maybe not useful for this concrete problem but interesting on its own) There's a theorem of Schur (see here Theorem 1) that gives a sufficient condition for a polynomial to be real rooted. I want to know if there's any kind of generalization in the case one changes the basis $\{1,x,x^2,\ldots\}$ for example with $\{\binom{x}{0},\binom{x}{1},\binom{x}{2},\ldots\}$.