# Real rootedness of a polynomial

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?

• Your observation follows trivially from the properties of zeroes of Jacobi polynomials.
– Nemo
Jan 30, 2020 at 18:15
• @Nemo, That doesn't seem completely obvious enough to say it is trivial. Can you maybe expand on your reasoning? Jan 30, 2020 at 18:18
• 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. Jan 30, 2020 at 18:52

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
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.
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).