Let $P_n(x)$ be a sequence of polynomials $\deg P_n= \deg P_{n-1}+1.$ Suppose that $$P_n(x)=((a_nn-b_n)x+b_n)P_{n-1}(x)+x(\alpha_n-\alpha_nx)P'_{n-1}(x)$$
I'm having trouble understanding the following statement:
$P_n(x)$ has only real roots if and only if the leading coefficient $\alpha_n \geq 0$.
From what I searched, this type of polynomials $P_n(x)$ forms something called Sturm's sequence, but I couldn't find when this type sequence has only real roots.
