Thread: orthogonal polynomials and the recurrence relation

1. orthogonal polynomials and the recurrence relation

Let $\langle\cdot,\cdot\rangle$ be an inner product on the real vector space of polynomials, and consider the set $\{p_0,\cdots,p_n,\cdots\}$ of monic orthogonal polynomials defined recursively by

$p_0(x)=1$

$p_1(x)=x-a_1$ and

$p_n(x)=(x-a_n)p_{n-1}-b_np_{n-2}(x)$ for $n\geq 2$,

where $a_n=\frac{\langle xp_{n-1},p_{n-1}\rangle}{\langle p_{n-1},p_{n-1}\rangle}$, and

$b_n=\frac{\langle xp_{n-1},p_{n-2}\rangle}{\langle p_{n-2},p_{n-2}\rangle}$.

Show that $\langle xp_{n-1},p_{n-2}\rangle=\langle p_{n-1},xp_{n-2}\rangle$.
I've been wrestling with this for several days, but I can't crack it. I don't think it's supposed to be difficult though. I suspect I'm just missing something obvious.

Any help would be much appreciated!

2. Re: orthogonal polynomials and the recurrence relation

Just a guess but it could be a special polynomial such as Legendre, Chebshev, etc. Look into those, because I don't remember what they look like.