Prove that the Dirac delta function can be presented in terms of the Legendre Polynomials

Question

I tried everything I could, but I can't find a derivation for this equation. Can I get some hints on how to approach to prove the following result? $$\delta(1-x)=\sum_{n=0}^{\infty}{\frac{(2n+1)P_n(x)}{2}}.$$

Answer 1

Just create a Fourier series using the orthogonality of the Legendre Polynomials. Set

$\delta (1-x)=\sum _{n=0}^{\infty } A_n\ P_n(x)$

Multiply both sides by $P_{n1}(x)$ and integrate.

$\int_{-1}^1 \delta (1-x) P_{n1}(x) \, dx=\sum _{n=0}^{\infty } A_n \int_{-1}^1 P_n(x) P_{{n1}}(x) \, dx$

$1$=$2$ $\frac{A_n \delta _{\text{n1},n}}{2 n+1}$

And solve for $A_n$.