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):

[math]\displaystyle{ \mathcal{K}_0(x; n) = 1, }[/math]

[math]\displaystyle{ \mathcal{K}_1(x; n) = -2x + n, }[/math]

[math]\displaystyle{ \mathcal{K}_2(x; n) = 2x^2 - 2nx + \binom{n}{2}, }[/math]

[math]\displaystyle{ \mathcal{K}_3(x; n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + \binom{n}{3}. }[/math]

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

[math]\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. }[/math]

Properties

The Kravchuk polynomial has the following alternative expressions:

[math]\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}. }[/math]

[math]\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}. }[/math]

Symmetry relations

For integers [math]\displaystyle{ i,k \ge 0 }[/math], we have that

[math]\displaystyle{ \begin{align} (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{align} }[/math]

Orthogonality relations

For non-negative integers r, s,

[math]\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}. }[/math]

Generating function

The generating series of Kravchuk polynomials is given as below. Here [math]\displaystyle{ z }[/math] is a formal variable.

[math]\displaystyle{ \begin{align} (1+(q-1)z)^{n-x}(1-z)^x &= \sum_{k=0}^\infty \mathcal{K}_k(x;n,q) {z^k}. \end{align} }[/math]

Three term recurrence

The Kravchuk polynomials satisfy the three-term recurrence relation

[math]\displaystyle{ \begin{align} 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{align} }[/math]

References

Kravchuk, M. (1929), "Sur une généralisation des polynomes d'Hermite." (in French), Comptes Rendus Mathématique 189: 620–622, 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 978-0-521-19225-5, http://dlmf.nist.gov/18.19
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 .
"Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", IEEE Transactions on Information Theory 41 (5): 1303–1321, 1995, doi:10.1109/18.412678 .
MacWilliams, F. J.; Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes, North-Holland, ISBN 0-444-85193-3, https://archive.org/details/theoryoferrorcor0000macw