Kravchuk polynomials

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Kravchuk polynomials or Krawtchouk polynomials are classical orthogonal polynomials associated with the binomial distribution, introduced by the Ukrainian mathematician Mikhail Kravchuk in 1929.[1]

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.

[edit] References

  1. ^ Sur une généralisation des polynomes d'Hermite. Note de M.Krawtchouk, C.R.Acad. Sci. 1929. T.189, No.17. P.620 - 622.
  • Nikiforov, A. F., Suslov, S. K. and Uvarov, V. B., "Classical Orthogonal Polynomials of a Discrete Variable". Springer-Verlag, Berlin-Heidelberg-New York, 1991.

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