Abramowitz formula for Gauss-Legendre’s weights

Question

I'm currently studying Gauss-Legendre quadrature formulas on the interval $[-1,1]$ and I see everywhere the following formula for the weights, where $x_1,…,x_n$ are the nodes and $p_n$ is the Legendre polynomial of degree $n$:
$$w_i=\frac{2}{(1-x_i^2)[p_n'(x_i)]^2} \quad \quad (1)$$
presented in: Abramowitz, Milton; Stegun, Irene Ann, (1972) "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", pp. 887.
The problem is that I don't understand how to get it from the general formula for the Gaussian weights:
$$w_i=\frac{a_n}{a_{n-1}}\frac{h_{n-1}}{p_n'(x_i)p_{n-1}(x_i)} \quad \quad (2)$$
where $h_{n-1}=\int_{-1}^1p_{n-1}(x)^2dx$.
I tried to use the recursive formula and the differential equation, considering the normalization $p_n(1)=1$, but I couldn't manage to get from $(2)$ to $(1)$. Any help would be greatly appreciated.

Answer 1

This webpage from Wolfram MathWorld connects the two formulas.