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.

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

• 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

• Krawtchouk Polynomials Home Page
• "Krawtchouk polynomial" at MathWorld