Rodrigues' formula

Remarks

That $q_n$ is a polynomial follows by induction: each derivative of $w \sigma^n$ is $w \sigma^{n-k}$ times a polynomial of degree at most $k$. This holds because $\sigma w' / w$ is a polynomial of degree at most $1$. For orthogonality, take $m < n$ and integrate $\int_I x^m\, \frac{d^n}{dx^n}(w \sigma^n)\, dx$ by parts $n$ times. The boundary terms vanish because $w \sigma^n$ and its first $n - 1$ derivatives vanish at the ends of $I$, and the final integrand contains $\frac{d^n}{dx^n} x^m = 0$. The usual normalisations are $P_n^{(\alpha,\beta)} = \frac{(-1)^n}{2^n n!}\, q_n$ (Jacobi), $L_n^{(\alpha)} = \frac{1}{n!}\, q_n$ (Laguerre), and $H_n = (-1)^n q_n$ (Hermite). Example: in the Hermite case with $n = 2$, $\frac{d^2}{dx^2} e^{-x^2} = (4x^2 - 2) e^{-x^2}$, so $q_2 = 4x^2 - 2 = H_2$. Example: in the Jacobi case with $\alpha = \beta = 0$, the formula gives the Legendre polynomials $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$.

Going deeper

For a student

Rodrigues' formula is the quickest way to compute norms, leading coefficients and generating functions of the classical families. With Cauchy's integral formula it gives contour integral representations, from which generating functions such as $\sum_n P_n(x)\, t^n = (1 - 2xt + t^2)^{-1/2}$ follow. The existence of such a formula characterises the classical families among orthogonal polynomials, together with Bochner's characterisation by differential equations. The polynomial $\sigma$ is the same one that appears in the second-order differential equation $\sigma y'' + \tau y' + \lambda_n y = 0$ satisfied by $q_n$.