Gauss-Jacobi Mechanical Quadrature

If $x_1<x_2<\ldots<x_n$ denote the zeros of $p_n(x)$, there exist Real Numbers $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that

$$\int_a^b \rho(x)\,d\alpha(x)=\lambda_1\rho(x_1)+\lambda_2\rho(x_2)+\ldots+\lambda_n\rho(x_n),$$

for an arbitrary Polynomial of order $2n-1$ and the $\lambda_n's$ are called Christoffel Numbers. The distribution $d\alpha(x)$ and the Integer $n$ uniquely determine these numbers $\lambda_\nu$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 47, 1975.