 Kravchuk polynomials

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:For binary linear codes, $q=2$. Expanding the factors gives $K_s(t;n,2)={\sum }_j(-1{)}^j(\genfrac{}{}{0ex}{}{t}{j}\left)\right(\genfrac{}{}{0ex}{}{n-t}{s-j})$, where binomial coefficients outside their usual integer range are zero. Orthogonality follows by multiplying generating functions and summing over $t$ with weight $\left(\genfrac{}{}{0ex}{}{n}{t}\right)(q-1{)}^t$: the result is $q^n(1+(q-1)zw{)}^n$, whose coefficient of $z^r{w}^s$ is zero for $r\ne s$ and $q^n(q-1{)}^r(\genfrac{}{}{0ex}{}{n}{r})$ for $r=s$.

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.