# Legendre polynomial property

1. May 5, 2007

### neelakash

1. The problem statement, all variables and given/known data

I am to prove that P_n(-x)=(-1)^n*P_n(x)

And, P'_n(-x)=(-1)^(n+1)*P'_n(x)

2. Relevant equations

3. The attempt at a solution

I know that whether a Legendre Polynomial is an even or odd function depends on its degree.It follows directly from the solution of Legendre differential equation.But,to prove these properties I am getting stuck.

2. May 5, 2007

3. May 5, 2007

### neelakash

I referred to those webpages earlier.They contain results but no derivation.

How can I use (-x)^a=(-1)^a*(x)^a within a series?

4. May 6, 2007

### siddharth

As Astronuc suggested, substitute as (-x)a for each term in the series. What can you say about (-1)n-2m when m varies?

5. May 6, 2007

### neelakash

One should exploit the property: P_n(x) even and odd according as n is even and odd.

6. May 6, 2007

### tnho

The statement you are going to prove can be seen readily if you look at the Ridrigues' formula of Lengreda Polynomial, isn't it?

7. May 6, 2007

### neelakash

I am interested to know how the first formula can be derived from Rodrigues's formula.Is the Replacing x by -x should affect (d^l/dx^l)?If so,how?

Actually it appears...but not quite sure,how?

8. May 6, 2007

### tnho

just change x-> -x
then the Rodrigues's formula has d/d(-x)
which can be rewritten as dx/d(-x)*d/d(x) , like changing variables as usual
then d/d(-x) = -d/dx and d^n/d(-x)^n = (-1)^n d^n/dx^n

for the question how the Rodrigues's formula came from...
i have no idea at all...

9. May 6, 2007

### neelakash

I anticipated something like this.Thanks for clarification.