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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?

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    $\begingroup$ Your observation follows trivially from the properties of zeroes of Jacobi polynomials. $\endgroup$ – Nemo Jan 30 at 18:15
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    $\begingroup$ @Nemo, That doesn't seem completely obvious enough to say it is trivial. Can you maybe expand on your reasoning? $\endgroup$ – JoshuaZ Jan 30 at 18:18
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    $\begingroup$ Though the question already has answers, it might still be helpful to know that there is a handy criterion for the real-rootedness of a polynomial by checking the non-negativity of quadratic form that can be built from its coefficients. $\endgroup$ – Igor Khavkine Jan 30 at 18:52
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If you have two polynomials $f(x)=a_0+a_1x+\cdots a_mx^m$ and $g(x)=b_0+b_1x+\cdots+b_nx^n$, such that the roots of $f$ are all real, and the roots of $g$ are all real and of the same sign, then the Hadamard product $$f\circ g(x)=a_0b_0+a_1b_1x+a_2b_2x^2+\cdots$$ has all roots real. This was proven originally in

E. Malo, Note sur les équations algébriques dont toutes les racines sont réelles, Journal de Mathématiques Spéciales, (4), vol. 4 (1895)

One can make stronger statements, such as the result by Schur that says that $\sum k!a_kb_k x^k$ will only have real roots, under the same conditions. Schur's theorem combined with the fact that $\{1/k!\}_{k\geq 0}$ is a Polya frequency sequence, implies Malo's theorem.

I'm not sure what the best reference to learn the theory of real rooted polynomials, and the associated operations that preserve real rootedness is. One textbook I know that discusses some of these classical results is Marden's "Geometry of Polynomials".

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According to the representation for Jacobi polynomials https://en.wikipedia.org/wiki/Jacobi_polynomials#Alternate_expression_for_real_argument $$ P^{(0,n-m)}_m(x)=\sum_{j=0}^m \binom{m}{j}\binom{n}{j}\left(\frac{x-1}{2}\right)^{j}\left(\frac{x+1}{2}\right)^{m-j} $$ OPs polynomial equals $$ P_{m,n}(t)=(1-t)^mP^{(0,n-m)}_m\left(\frac{1+t}{1-t}\right). $$ Since zeroes of Jacobi polynomials are real valued, all roots of the polynomial $P_{m,n}(t)$ are also real valued (see D.Dominici, S.J.Johnston, K.Jordaan, Real zeros of hypergeometric polynomials).

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