# .

Kravchuk polynomials or Krawtchouk polynomials (and several other transliterations of the Ukrainian Кравчу́к) are discrete orthogonal polynomials associated with the binomial distribution, introduced by Krawtchouk (1929). The first few polynomials are:

$$\mathcal{K}_0(x, n) = 1$$
$$\mathcal{K}_1(x, n) = -2x + n$$
$$\mathcal{K}_2(x, n) = 2x^2 - 2nx + {n\choose 2}$$
$$\mathcal{K}_3(x, n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + {n \choose 3}$$.

The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
References

Krawtchouk, M. (1929), "Sur une généralisation des polynomes d'Hermite." (in French), Comptes Rendus Mathematique 189: 620–622, ISSN 1631-073X, 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248
Nikiforov, A. F., Suslov, S. K. and Uvarov, V. B., "Classical Orthogonal Polynomials of a Discrete Variable". Springer-Verlag, Berlin-Heidelberg-New York, 1991.
Levenshtein, V.I. "Krawtchouk polynomials and universal bounds for codes and designs in Hamming space," IEEE Transactions on Information Theory, vol. 41, number 5, pp. 1303–1321, 1995.