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):

${\displaystyle {\mathcal {K}}_{0}(x;n)=1,}$

${\displaystyle {\mathcal {K}}_{1}(x;n)=-2x+n,}$

${\displaystyle {\mathcal {K}}_{2}(x;n)=2x^{2}-2nx+{\binom {n}{2}},}$

${\displaystyle {\mathcal {K}}_{3}(x;n)=-{\frac {4}{3}}x^{3}+2nx^{2}-(n^{2}-n+{\frac {2}{3}})x+{\binom {n}{3}}.}$

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