Why is the sign of terms in orthogonal polynomials always alternating?

Question

I checked some expressions of orthogonal polynomials, e.g. Laguerre Polymonial, Legendre Polynomial, Hermite Polynomials, etc. And the sign of terms in them are always alternating. For example, $$H_8(x) = 256 x^8-3584 x^6+13440 x^4-13440 x^2+1680$$ The sign of each term is $(+~-~+~-~+)$. This seems to be true for every polynomials. Is there an explanation for this?

Answer 1

I think this follows from the fact that all these families satisfy a three-term recurrence relation $$ P_n(x) = (x-\alpha_{n-1}) P_{n-1}(x)-\beta_{n-2} P_{n-2}(x) $$

Answer 2

Once we have a Rodrigues-like formula (encoding a particular choice of an orthogonal base) the alternating signs are a pretty straightforward consequence. For instance, in the Legendre case $$ P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n.\tag{1}$$ The binomial expansion of $(x^2-1)^n$ has alternating signs: trivial. The operator $\frac{d^n}{dx^n}$ does not change that, neither it does the multiplication by $\frac{1}{2^n n!}$.