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

$\sum_{j=0}^n \frac{f_j(x) f_j(y)}{h_j} = \frac{k_n}{h_n k_{n%2B1}} \frac{f_n(y) f_{n%2B1}(x) - f_{n%2B1}(y) f_n(x)}{x - y}$

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

References

Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, ISBN 978-0-521-62321-6; 978-0-521-78988-2, MR 1688958
Christoffel, E. B. (1858), "Über die Gaußische Quadratur und eine Verallgemeinerung derselben." (in German), Journal für Reine und Angewandte Mathematik 55: 61–82, doi:10.1515/crll.1858.55.61, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002150239
Darboux, Gaston (1878), "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série" (in French), Journal de Mathématiques Pures et Appliquées 4: 5–56, 377–416, JFM 10.0279.01