Kravchuk polynomials
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian name "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by Mikhail Kravchuk (1929). The first few polynomials are (for q=2):
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
Definition
For any prime power q and positive integer n, define the Kravchuk polynomial
Properties
The Kravchuk polynomial has the following alternative expressions:
Orthogonality relations
For nonnegative integers r, s,
See also
References
- Kravchuk, M. (1929), "Sur une généralisation des polynomes d'Hermite.", Comptes Rendus Mathematique (in French), 189: 620–622, JFM 55.0799.01
- 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.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B. (1991), Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Berlin: Springer-Verlag, ISBN 3-540-51123-7, MR 1149380.
- Levenshtein, Vladimir I. (1995), "Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", IEEE Transactions on Information Theory, 41 (5): 1303–1321, doi:10.1109/18.412678, MR 1366326.
- MacWilliams, F. J.; Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes, North-Holland, ISBN 0-444-85193-3
External links
Wikimedia Commons has media related to Kravchuk polynomials. |
This article is issued from Wikipedia - version of the 9/9/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.