An elementary proof of Rodrigues' formula for Legendre polynomials (which is usually done via the orthogonality properties of the polynomials). If we define the polynomials by the classical generating function,

$$ \frac{1}{\sqrt{1-2z_{0}w+w^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(z_{0}\right)w^{n} $$

and consider the solution $z(w)$ of $$w=f(z)=2\frac{z-z_{0}}{z^{2}-1}$$ fixed by $z(0)=z_0$, then the direct application of the Lagrange's formula (formulated for the solution $z(w)$ of the equation $w=f(z)$ with $w_0=f(z_0)$) $$\frac{\Phi\left(z\left(w\right)\right)}{1-\left(w-w_{0}\right)\left.\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{z-z_{0}}{f(z)-w_{0}}\right)\right|_{z=z(w)}}=\sum_{n=0}^{\infty}c_{n}\left(w-w_{0}\right)^{n},\quad c_{n}=\frac{1}{n!}\left.\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\left[\Phi\left(z\right)\left(\frac{z-z_{0}}{f(z)-w_{0}}\right)^{n}\right]\right|_{z=z_{0}}$$ to $f(z)$, $w_0=f(z_0)=0$ and $\Phi(z)\equiv 1$ gives $$P_{n}(z)=\frac{1}{2^{n}n!}\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\left[\left(z^{2}-1\right)^{n}\right].$$