One of the most important properties of Orthogonal polynomials is the ff.
Theorem: If $\{p_n (x)\}_{n=0}^\infty$ is a sequence of orthogonal polynomials on the interval $(a,b)$ with respect to the weight function $w(x)$, then the polynomial $p_n (x)$ has exactly $n$ real simple zeros in the interval $(a,b)$.
Can you prove it? Please do it for me. The proof included in aw.twi.tudelft.nl /~koekoek/documents/wi4006/orthopoly.pdf is in shortcut form. I cant understand it.