Question

What is Legendre coefficients?

Answer

Legendre coefficients is a general solution of Legendre's equation. $\left(1-x^2\right)\frac{d^{2}z}{{\mathrm{dx}}^2}-2x\frac{\mathrm{dz}}{\mathrm{dx}}+m\left(m+1\right)z=0$
$\mathrm{where\; m\; is\; a\; positive\; integer,\; is}$
$\mathrm{z }\mathrm{=}A{P}_m\left(x\right)+B{Q}_m\left(x\right)$
$P_m\left(x\right)=\frac{\left(\mathrm{2m}-1\right)\left(\mathrm{2m}-3\right)\mathrm{....1}}{\mathrm{m!}}$
$\left\{x^m\frac{m}{2\left(\mathrm{2m}\right)}+\frac{m\left(m-1\right)\left(m-2\right)\left(m-3\right)}{2.4\left(\mathrm{2m}-1\right)\left(\mathrm{2m}-3\right)}{x}^{\mathrm{m-4}}\mathrm{...}\right\}$