Legendre Type polynomial

**HTale** · December 21st 2008, 05:24 AM

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

**Mush** · December 21st 2008, 08:34 AM

Originally Posted by HTale

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?

**HTale** · December 21st 2008, 09:10 AM

Originally Posted by Mush

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.

**Opalg** · December 21st 2008, 09:48 AM

Originally Posted by HTale

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.