Christoffel Number

One of the quantities $\lambda_i$ appearing in the Gauss-Jacobi Mechanical Quadrature. They satisfy

$\begin{displaymath} \lambda_1+\lambda_2+\ldots+\lambda_n=\int_a^b d\alpha(x)=\alpha(b)-\alpha(a) \end{displaymath}$

(1)

and are given by

$\displaystyle \lambda_\nu$	$\textstyle =$	$\displaystyle \int_a^b \left[{p_n(x)\over p_n'(x_\nu)(x-x_\nu)}\right]^2\,d\alpha(x)$	(2)
$\displaystyle \lambda_\nu$	$\textstyle =$	$\displaystyle -{k_{n+1}\over k_n}{1\over p_{n+1}(x_\nu)p_n'(x_\nu)}$	(3)
	$\textstyle =$	$\displaystyle {k_n\over k_{n-1}} {1\over p_{n-1}(x_\nu)P_n'(x_\nu)}$	(4)
$\displaystyle (\lambda_\nu)^{-1}$	$\textstyle =$	$\displaystyle [p_0(x_\nu)]^2+\ldots+[p_n(x_\nu)]^2,$	(5)

where

is the higher Coefficient of

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 47-48, 1975.