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Thread: orthogonal polynomials and the recurrence relation

  1. #1
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    orthogonal polynomials and the recurrence relation

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

    $\displaystyle p_0(x)=1$

    $\displaystyle p_1(x)=x-a_1$ and

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

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

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

    Show that $\displaystyle \langle xp_{n-1},p_{n-2}\rangle=\langle p_{n-1},xp_{n-2}\rangle$.
    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!
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  2. #2
    MHF Contributor
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    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.
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