Question

What is Legendre coefficients?

Answer

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

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