Timeline for Orthogonal basis of polynomials?

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Sep 13, 2021 at 15:33	comment	added	Hexhist		The 3 term recurrence relation does not immediately imply that the $p_n$ do not have a common root. In fact, if $p_n$ and $p_{n-1}$ have the same root, the recursion implies that $p_{n+1}$ has also the same root. The reason that common roots are not allowed is more subtle. In any event, I do consider that Fernando’s question is a very interesting one, and deserves a more informative answer, if possible.
Oct 25, 2018 at 23:28	comment	added	Chris Godsil		Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics” discusses it at lengh too ( if you’ll forgive the plug).
Oct 25, 2018 at 23:04	comment	added	user41593		Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
Oct 25, 2018 at 22:38	vote	accept	fernando
Oct 25, 2018 at 22:38	vote	accept	fernando
Oct 25, 2018 at 22:38
Oct 25, 2018 at 22:35	comment	added	fernando		Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = \int \frac{dz}{z} P_n(x) P_{-m}(x)$, where the integral is a contour integral around zero.
Oct 25, 2018 at 22:34	vote	accept	fernando
Oct 25, 2018 at 22:38
Oct 25, 2018 at 22:29	history	answered	Chris Godsil	CC BY-SA 4.0