Using the generating function for the Legendre polynomials

Printable View

May 17th 2011, 08:37 AM
garunas

Using the generating function for the Legendre polynomials

Using the identity:

$\sum_{n=0}^\infty P_{n}(x)r^n = \frac{1}{\sqrt {1-2xr+r^2} }$ with:

$x \in [-1,1], r \in (-1,1)$

Derive an expression for $P_{2}(x)$ the legendre polynomial of degree 2.

Any help here will be amazing as I absolutely haven't got a clue!

Also from this it also asks to evaluate $P_{3}'(0)$ but i take that if you can find $P_{2}(x)$ then you can also find $P_{2}'(0)$ and use the same method for $P_{3}(x)$

Thank you
May 17th 2011, 09:02 AM
TheEmptySet

Quote:

Originally Posted by garunas

Using the identity:

$\sum_{n=0}^\infty P_{n}(x)r^n = \frac{1}{\sqrt {1-2xr+r^2} }$ with:

$x \in [-1,1], r \in (-1,1)$

Derive an expression for $P_{2}(x)$ the legendre polynomial of degree 2.

Any help here will be amazing as I absolutely haven't got a clue!

Also from this it also asks to evaluate $P_{3}'(0)$ but i take that if you can find $P_{2}(x)$ then you can also find $P_{2}'(0)$ and use the same method for $P_{3}(x)$

Thank you

you need to expand the right hand side in a Taylor series around zero

so let

$f(r)=(1-2xr+r^2)^{-\frac{1}{2}}$

Now you get

$1+f'(0)r+\frac{f''(0)}{2!}r^2+\frac{f'''(0)}{3!}r^ 3+...$

Now just equate coefficients this gives

$P_2(x)r^2=\frac{f''(0)}{2!}r^2 \implies P_2(x)=\frac{f''(0)}{2!}$

Can you finish from here?
May 17th 2011, 09:41 AM
garunas

...i can sure finish from here. Thanks a lot! :)
May 17th 2011, 10:28 AM
Opalg

Quote:

Originally Posted by garunas

Using the identity:

$\sum_{n=0}^\infty P_{n}(x)r^n = \frac{1}{\sqrt {1-2xr+r^2} }$ with:

$x \in [-1,1], r \in (-1,1)$

Derive an expression for $P_{2}(x)$ the legendre polynomial of degree 2.

Any help here will be amazing as I absolutely haven't got a clue!

Also from this it also asks to evaluate $P_{3}'(0)$ but i take that if you can find $P_{2}(x)$ then you can also find $P_{2}'(0)$ and use the same method for $P_{3}(x)$

$\frac{1}{\sqrt {1-2xr+r^2} } = \bigl(1-(2xr-r^2)\bigr)^{-1/2}$ . Use the first few terms of the binomial expansion of $(1-t)^{-1/2}$ (with $t=2xr-r^2$ ) to find the coefficient of $r^2$ .

You could get $P_3'(0)$ by the same method. Alternatively, it might be slightly easier to start by differentiating $\frac{1}{\sqrt {1-2xr+r^2} }$ with respect to x, then put x=0, and finally use the binomial series to find the coefficient of $r^3$ .