1. ## Legendre Type polynomial

Hi guys,

A quick question, the following polynomial

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

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?

HTale

2. Originally Posted by HTale
Hi guys,

A quick question, the following polynomial

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

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?

HTale
That depends. What kind of Legendre polynomial are you used to?

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

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

Has this been shifted in some way, to produce the above one? It's really close to the shifted polynomial, but not quite.

4. Originally Posted by HTale
Hi guys,

A quick question, the following polynomial

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

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.

5. Originally Posted by Opalg
So, the shift is $\displaystyle 1-2x$ instead?