Kravchuk polynomials
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by Mykhailo 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:
Symmetry relations
Orthogonality relations
For non-negative integers r, s,
Generating function
The generating series of Kravchuk polynomials is given as below. Here is a formal variable.
See also
References
- Kravchuk, M. (1929), "Sur une généralisation des polynomes d'Hermite.", Comptes Rendus Mathématique (in French), 189: 620–622, JFM 55.0799.01CS1 maint: unrecognized language (link)
- 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. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, 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
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