Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian name "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by Mikhail Kravchuk (1929). The first few polynomials are (for q=2):

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

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

Definition

For any prime power q and positive integer n, define the Kravchuk polynomial

$\mathcal{K}_k(x; n) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n.$

Properties

The Kravchuk polynomial has following alternative expressions:

$\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}.$
$\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}.$

Orthogonality relations

For nonnegative integers r, s,

$\sum_{i=0}^n\binom{n}{i}(q-1)^i\mathcal{K}_r(i; n)\mathcal{K}_s(i; n) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}.$