How to Prove Properties of Legendre Polynomials?

[neelakash]

Homework Statement

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

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

Homework Equations

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.

Can anyone help me to start with these?

 

[Astronuc]

Have you tried using the series expression for Legendre polynomials and then using (-x)^a = (-1)^a(x)^a?

See - http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html

http://mathworld.wolfram.com/LegendrePolynomial.html

http://en.wikipedia.org/wiki/Legendre_polynomials


Also try that approach with differentiation.

 

[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?

 

[siddharth]

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?

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

 

[neelakash]

OK,I already got it.
One should exploit the property: P_n(x) even and odd according as n is even and odd.

 

[tnho]

Homework Statement

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

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

Homework Equations

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.

Can anyone help me to start with these?

OK,I already got it.
One should exploit the property: P_n(x) even and odd according as n is even and odd.

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

 

[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?

 

[tnho]

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?

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...

 

[neelakash]

I anticipated something like this.Thanks for clarification.