Let α(x) be a step function with the jump j(x) = (N x) p^x q^(N - x) at x = 0, 1, ..., N, where p>0, q>0, and p + q = 1. Then the Krawtchouk polynomial is defined by k_n^(p)(x, N) | = | sum_(ν = 0)^n (-1)^(n - ν)(N - x n - ν)(x ν) p^(n - ν) q^ν, | = | (-1)^n(N n) p^n _2 F_1(-n, - x; - N;1/p) | = | ((-1)^n p^n)/(n!) (Γ(N - x + 1))/(Γ(N - x - n + 1))×_2 F_1(-n, - x;N - x - n + 1;(p - 1)/p). for n = 0, 1, ..., N.

Meixner polynomial of the first kind | orthogonal polynomials