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Math Help - Legendre Type polynomial

  1. #1
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    Legendre Type polynomial

    Hi guys,

    A quick question, the following polynomial

    \displaystyle n!P_n(x) = \displaystyle \{\frac{d}{dx}\}^nx^n(1-x)^n

    was considered a ""Legendre Type" polynomial. It's certainly not a Legendre polynomial I'm used to. Is there a difference? If so, what is it?

    Thanks in advance,

    HTale
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  2. #2
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    Quote Originally Posted by HTale View Post
    Hi guys,

    A quick question, the following polynomial

    \displaystyle n!P_n(x) = \displaystyle \{\frac{d}{dx}\}^nx^n(1-x)^n

    was considered a ""Legendre Type" polynomial. It's certainly not a Legendre polynomial I'm used to. Is there a difference? If so, what is it?

    Thanks in advance,

    HTale
    That depends. What kind of Legendre polynomial are you used to?
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  3. #3
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    Quote Originally Posted by Mush View Post
    That depends. What kind of Legendre polynomial are you used to?
    Ones of the form

    \displaystyle 2^nn!P_n(x) = \{\frac{d}{dx}\}^n[(x^2-1)^n]

    Has this been shifted in some way, to produce the above one? It's really close to the shifted polynomial, but not quite.
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  4. #4
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    Quote Originally Posted by HTale View Post
    Hi guys,

    A quick question, the following polynomial

    \displaystyle n!P_n(x) = \displaystyle \{\frac{d}{dx}\}^nx^n(1-x)^n

    was considered a ""Legendre Type" polynomial. It's certainly not a Legendre polynomial I'm used to. Is there a difference? If so, what is it?
    These are shifted Legendre polynomials.
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  5. #5
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    Quote Originally Posted by Opalg View Post
    So, the shift is 1-2x instead?
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