Integral of a product of othogonal polynomials over the domain

Question

Let $\phi_i (x)$ with $x \in I \subseteq \mathbb{R}$ be an i-th order orthogonal polynomials with respect to the weight function $w(x)$.

Is the following integral finite for any othogonal polynomial and every $n$? Why?

$\int_I \prod_{i=0}^{n} \phi_i (x) w(x) dx $

for $n \in \mathbb{N}^{*}$.

Answer 1

Let me just prove the following \begin{align} \int^\infty_0 P_n(x)e^{-x}\ dx <\infty \end{align} for any polynomial $P_n(x)$. It suffices to prove the integral for $P_n(x)= x^n$. Observe \begin{align} \int^\infty_0 x^ne^{-x}\ dx =&\ \int^\infty_0 (-1)^n\frac{d^n}{dt^n}e^{-tx}\bigg|_{t=1}\ dx = (-1)^n\frac{d^n}{dt^n}\int^\infty_0 e^{-tx}\ dx\\ =&\ (-1)^{n} \frac{d^n}{dt^n} \left(\frac{1}{t}\right)\bigg|_{t=1} = n!. \end{align}