How to show that any polynomial of degree $\le n$ can be written as a linear combination of the $L_j(x)$ for $j = 0,1,2,...,n$
$P_n(x)=a_0L_0(x)+a_1L_1(x)+a_2L_2(x)+\dots+a_nL_n(x)$
ANSWER :: the legendre polynomials of degree ≤ n, are linearly independent, and thus form a basis for all polynomials of degree ≤ n. therefore, every polynomial of degree ≤ n can be written as a linear combination
thanks. :-)
A little mixup in notation there. $P_n(x)$ is normally denoting the $n^{th}$ Legendre polynomial, and $L_n(x)$ the $n^{th}$ Laguerre polynomial.
Both $\{ P_n| n \in \mathbb{N} \}$ and $\{ L_n| n \in \mathbb{N} \}$ form sets of orthogonal functions, which means that when taking an inner product of two of its members which are different, then the result is zero. Here the inner product is some integral of the product of the functions.
See for example Legendre Polynomial Orthogonality and Size and Legendre polynomials, Laguerre polynomials: Basic concept.
Any function on the relevant interval ($[-1,1]$ for Legendre polynomials, $[0,\infty]$ for Laguerre polynomials) can then be expanded into a series with terms $a_nP_n(x)$ or $a_nL_n(x)$ instead of the 'normal' $a_nx^n$.
In particular, any polynomial $P(x)$ can be expanded on $[-1,1]$ as $\sum_{j=0}^{\infty} a_jP_j(x)$ with appropriate coefficients $a_j$.
See Using Legendre polynomial to approximate any polynomial.