# Legendre Polynomials Proof

a.) Prove that the even (odd) degree Legendre polynomials are even (odd) functions of t.

b.) Prove that if $p(t) = p(-t)$ is an even polynomial, then all the odd order coefficents $c_{2j+1} = 0$ in its Legendre expansion $p(t) = c_0q_0(t) + ... + c_nq_n(t)$ vanish.

Is "a" similar to proving that Legendre polynomials of even and odd degree are orthogonal and for "b" I do not know how to do.

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a) $$P_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n}\left[(x^2-1)^n \right].$$ Here $(x^2-1)^n$ is an even polynomial. Taking the $n$-th derivative the general terms have the form $a_k x^{2k-n}$ which is even iff $n$ is even, and odd iff $n$ is odd.
b) Writing out the Legendre-Fourier coefficients by integral on $[-1,1]$, then the integrand $p(t)P_{2n+1}(t)$ is an odd function so the value of the integral is $0$. (The situation is similar to the classical trigonometric Fourier series case on the interval $[-\pi,\pi]$.