Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials (and several other transliterations of the Ukrainian Кравчу́к) are discrete orthogonal polynomials associated with the binomial distribution, introduced by Krawtchouk (1929). The first few polynomials are:

$\mathcal{K}_0(x, n) = 1$
$\mathcal{K}_1(x, n) = -2x %2B n$
$\mathcal{K}_2(x, n) = 2x^2 - 2nx %2B {n\choose 2}$
$\mathcal{K}_3(x, n) = -\frac{4}{3}x^3 %2B 2nx^2 - (n^2 - n %2B \frac{2}{3})x %2B {n \choose 3}.$

The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.

References

Krawtchouk, M. (1929), "Sur une généralisation des polynomes d'Hermite." (in French), Comptes Rendus Mathématique. Académie des Sciences. Paris 189: 620–622, ISSN 1631-073X, JFM 55.0799.01, http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248, http://dlmf.nist.gov/18.19
Nikiforov, A. F., Suslov, S. K. and Uvarov, V. B., "Classical Orthogonal Polynomials of a Discrete Variable". Springer-Verlag, Berlin-Heidelberg-New York, 1991.
Levenshtein, V.I. "Krawtchouk polynomials and universal bounds for codes and designs in Hamming space," IEEE Transactions on Information Theory, vol. 41, number 5, pp. 1303–1321, 1995.

Kravchuk polynomials

References

External links