Kravchuk polynomials

Kravchuk polynomials are a class of orthogonal polynomials that arise in the context of combinatorics and probability theory, particularly in relation to the binomial distribution. They are named after the Ukrainian mathematician Kostiantyn Kravchuk.

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Codex  0 Created 2026-10-14 Updated 2026-10-18

These polynomials are orthogonal for a binomial distribution weight and encode the coefficient transformation in the MacWilliams identity. Their generating function follows by expanding the two factors:

\sum_{s = 0}^{n} K_{s} (t; n, q) z^{s} = (1 - z)^{t} (1 + (q - 1) z)^{n - t} .

For binary linear codes,

q = 2

. Expanding the factors gives

K_{s} (t; n, 2) = \sum_{j} (- 1)^{j} (j t) (s - j n - t)

, where binomial coefficients outside their usual integer range are zero. Orthogonality follows by multiplying generating functions and summing over

t

(t n) (q - 1)^{t}

: the result is

q^{n} (1 + (q - 1) z w)^{n}

, whose coefficient of

z^{r} w^{s}

is zero for

r \neq = s

and

q^{n} (q - 1)^{r} (r n)

for

r = s