Parity relation for Legendre polynomials

Question

Can anybody please tell me the parity relation for $(-1)^{-n} P_n(x)$. Note that $P_n(x)$ is the $n$th Legendre polynomial.

Answer 1

Legendre polynomial $P_n(x)$ is given explicitly by Rodrigues' formula

$$ P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}\left((x^2 - 1)^n\right). $$

Note that $(x^2 - 1)^n$ is an even function for every integer $n$. Also, recall that the derivative of an even function is odd and the derivative of an odd function is even (this follows easily from the chain rule). We conclude that $P_n(x)$ is even when $n$ is even and $P_n(x)$ is odd when $n$ is odd.

Also, note that constant factors such as $(-1)^{-n}$ do not influence parity.