Christoffel–Darboux formula

In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878). It states that

$\sum_{j=0}^n \frac{f_j(x) f_j(y)}{h_j} = \frac{k_n}{h_n k_{n+1}} \frac{f_n(y) f_{n+1}(x) - f_{n+1}(y) f_n(x)}{x - y}$

where f_j(x) is the jth term of a set of orthogonal polynomials of squared norm h_j and leading coefficient k_j.

See also

• Turán's inequalities
• Sturm Chain