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):
K 0 ( x ; n ) = 1 , {\displaystyle {\mathcal {K}}_{0}(x;n)=1,} K 1 ( x ; n ) = − 2 x + n , {\displaystyle {\mathcal {K}}_{1}(x;n)=-2x+n,} K 2 ( x ; n ) = 2 x 2 − 2 n x + ( n 2 ) , {\displaystyle {\mathcal {K}}_{2}(x;n)=2x^{2}-2nx+{\binom {n}{2}},} K 3 ( x ; n ) = − 4 3 x 3 + 2 n x 2 − ( n 2 − n + 2 3 ) x + ( n 3 ) . {\displaystyle {\mathcal {K}}_{3}(x;n)=-{\frac {4}{3}}x^{3}+2nx^{2}-(n^{2}-n+{\frac {2}{3}})x+{\binom {n}{3}}.} The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
Definition [ edit ] For any prime power q and positive integer n , define the Kravchuk polynomial
K k ( x ; n , q ) = K k ( x ) = ∑ j = 0 k ( − 1 ) j ( q − 1 ) k − j ( x j ) ( n − x k − j ) , k = 0 , 1 , … , n . {\displaystyle {\mathcal {K}}_{k}(x;n,q)={\mathcal {K}}_{k}(x)=\sum _{j=0}^{k}(-1)^{j}(q-1)^{k-j}{\binom {x}{j}}{\binom {n-x}{k-j}},\quad k=0,1,\ldots ,n.} Properties [ edit ] The Kravchuk polynomial has the following alternative expressions:
K k ( x ; n , q ) = ∑ j = 0 k ( − q ) j ( q − 1 ) k − j ( n − j k − j ) ( x j ) . {\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-q)^{j}(q-1)^{k-j}{\binom {n-j}{k-j}}{\binom {x}{j}}.} K k ( x ; n , q ) = ∑ j = 0 k ( − 1 ) j q k − j ( n − k + j j ) ( n − x k − j ) . {\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-1)^{j}q^{k-j}{\binom {n-k+j}{j}}{\binom {n-x}{k-j}}.} Symmetry relations [ edit ] For integers i , k ≥ 0 {\displaystyle i,k\geq 0} , we have that
( q − 1 ) i ( n i ) K k ( i ; n , q ) = ( q − 1 ) k ( n k ) K i ( k ; n , q ) . {\displaystyle {\begin{aligned}(q-1)^{i}{n \choose i}{\mathcal {K}}_{k}(i;n,q)=(q-1)^{k}{n \choose k}{\mathcal {K}}_{i}(k;n,q).\end{aligned}}} Orthogonality relations [ edit ] For non-negative integers r , s ,
∑ i = 0 n ( n i ) ( q − 1 ) i K r ( i ; n , q ) K s ( i ; n , q ) = q n ( q − 1 ) r ( n r ) δ r , s . {\displaystyle \sum _{i=0}^{n}{\binom {n}{i}}(q-1)^{i}{\mathcal {K}}_{r}(i;n,q){\mathcal {K}}_{s}(i;n,q)=q^{n}(q-1)^{r}{\binom {n}{r}}\delta _{r,s}.} Generating function [ edit ] The generating series of Kravchuk polynomials is given as below. Here z {\displaystyle z} is a formal variable.
( 1 + ( q − 1 ) z ) n − x ( 1 − z ) x = ∑ k = 0 ∞ K k ( x ; n , q ) z k . {\displaystyle {\begin{aligned}(1+(q-1)z)^{n-x}(1-z)^{x}&=\sum _{k=0}^{\infty }{\mathcal {K}}_{k}(x;n,q){z^{k}}.\end{aligned}}} Three term recurrence [ edit ] The Kravchuk polynomials satisfy the three-term recurrence relation
x K k ( x ; n , q ) = − q ( n − k ) K k + 1 ( x ; n , q ) + ( q ( n − k ) + k ( 1 − q ) ) K k ( x ; n , q ) − k ( 1 − q ) K k − 1 ( x ; n , q ) . {\displaystyle {\begin{aligned}x{\mathcal {K}}_{k}(x;n,q)=-q(n-k){\mathcal {K}}_{k+1}(x;n,q)+(q(n-k)+k(1-q)){\mathcal {K}}_{k}(x;n,q)-k(1-q){\mathcal {K}}_{k-1}(x;n,q).\end{aligned}}}
See also [ edit ] References [ 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 External links [ edit ]