The Kravchuk polynomial has the following alternative expressions: :\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}. :\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}. Note that there is more that merely recombination of material from the two binomial coefficients separating these from the above definition. In these formulae, only one term of the sum has degree k , whereas in the definition all terms have degree k .
Symmetry relations For integers i,k \ge 0, we have that :\begin{align} (q-1)^{i} {n \choose i} \mathcal{K}_k(i;n,q) = (q-1)^{k}{n \choose k} \mathcal{K}_i(k;n,q). \end{align}
Orthogonality relations For non-negative integers
r,
s, :\sum_{i=0}^n\binom{n}{i}(q-1)^i\mathcal{K}_r(i; n,q)\mathcal{K}_s(i; n,q) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}.
Generating function The
generating series of Kravchuk polynomials is given as below. Here z is a formal variable. :\begin{align} (1+(q-1)z)^{n-x}(1-z)^x &= \sum_{k=0}^\infty \mathcal{K}_k(x;n,q) {z^k}. \end{align}
Three term recurrence The Kravchuk polynomials satisfy the three-term recurrence relation :\begin{align} x \mathcal{K}_k(x;n,q) = - q(n-k) \mathcal{K}_{k+1}(x;n,q) + (q(n-k) + k(1-q)) \mathcal{K}_{k}(x;n,q) - k(1-q)\mathcal{K}_{k-1}(x;n,q). \end{align} ==See also==