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
editFor any prime power q and positive integer n, define the Kravchuk polynomial
Properties
editThe Kravchuk polynomial has the following alternative expressions:
Symmetry relations
editFor integers , we have that
Orthogonality relations
editFor non-negative integers r, s,
Generating function
editThe generating series of Kravchuk polynomials is given as below. Here is a formal variable.
Three term recurrence
editThe Kravchuk polynomials satisfy the three-term recurrence relation
See also
editReferences
edit- Kravchuk, M. (1929), "Sur une généralisation des polynomes d'Hermite.", Comptes Rendus Mathématique (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. (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