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.

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, MR 1688958
Christoffel, E. B. (1858), "Über die Gaußische Quadratur und eine Verallgemeinerung derselben.", Journal für Reine und Angewandte Mathematik (in German), 55: 61–82, ISSN 0075-4102, doi:10.1515/crll.1858.55.61
Darboux, Gaston (1878), "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série", Journal de Mathématiques Pures et Appliquées (in French), 4: 5–56, 377–416, JFM 10.0279.01
Abramowitz, Milton; Stegun, Irene A. (1972), Handbook of Mathematical Functions, Dover Publications, Inc., New York, p. 785, Eq. 22.12.1
Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (2010), "NIST Handbook of Mathematical Functions", NIST Digital Library of Mathematical Functions, Cambridge University Press, p. 438, Eqs. 18.2.12 and 18.2.13, ISBN 978-0-521-19225-5 (Hardback, ISBN 978-0-521-14063-8 Paperback)

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