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

where fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj.

There is also a "confluent form" of this identity:

See also


  • 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 die Reine und Angewandte Mathematik (in German), 55: 61–82, doi:10.1515/crll.1858.55.61, ISSN 0075-4102
  • 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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