Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials are classical orthogonal polynomials associated with the binomial distribution, introduced by the Ukrainian mathematician Mikhail Kravchuk in 1929. [Sur une généralisation des polynomes d'Hermite. Note de M.Krawtchouk, C.R.Acad. Sci. 1929. T.189, No.17. P.620 - 622. ]

The first few polynomials are:
* $mathcal\{K\}\_0(x,\; n)\; =\; 1$
* $mathcal\{K\}\_1(x,\; n)\; =\; -2x\; +\; n$
* $mathcal\{K\}\_2(x,\; n)\; =\; 2x^2\; -\; 2nx\; +\; \{nchoose\; 2\}$
* $mathcal\{K\}\_3(x,\; n)\; =\; -frac\{4\}\{3\}x^3\; +\; 2nx^2\; -\; (n^2\; -\; n\; +\; frac\{2\}\{3\})x\; +\; \{n\; choose\; 3\}.$

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

References

*Nikiforov, A. F., Suslov, S. K. and Uvarov, V. B., "Classical Orthogonal Polynomials of a Discrete Variable". Springer-Verlag, Berlin-Heidelberg-New York, 1991.

External links

* [http://orthpol.narod.ru/ Krawtchouk Polynomials Home Page]
* [http://mathworld.wolfram.com/KrawtchoukPolynomial.html "Krawtchouk polynomial"] at MathWorld