Krawtchouk Polynomial

Let $\alpha(x)$ be a Step Function with the Jump

$$j(x)={N\choose x}p^xq^{N-x}$$ (1)

at $x=0$, 1, ..., $N$, where $p>0, q>0$, and $p+q=1$. Then

$$k_n^{(p)}(x)=\left[{{N\choose n}}\right]^{-1/2} (pq)^{-n/2}\... ...n (-1)^{n-\nu}{N-x\choose n-\nu}{x\choose\nu} p^{n-\nu} q^\nu,$$ (2)

for $n=0$, 1, ..., $N$. It has Weight Function

$$w={N! p^xq^{N-x}\over\Gamma(1+x)\Gamma(N+1-x)},$$ (3)

where $\Gamma(x)$ is the Gamma Function, Recurrence Relation

$$(n+1)k_{n+1}^{(p)}(x)+pq(N-n+1)k_{n-1}^{(p)}(x) = [x-n-(N-2)]k_n^{(p)}(x),$$ (4)

and squared norm

$${N!\over n!(N-n)!}(pq)^n.$$ (5)

It has the limit

$$\lim_{n\to\infty} \left({2\over Npq}\right)^{n/2} n! k_n^{(p)}(Np+\sqrt{2Npq}s)=H_n(s),$$ (6)

where $H_n(x)$ is a Hermite Polynomial, and is related to the Hypergeometric Function by

$k_n^{(p)}(x,N)=k_n^{(p)}(x,N)=(-1)^n{N\choose n}p^n\,{}_2F_1(-n, -x; -N; 1/p)$
${(-1)^n p^n\over n!}{\Gamma(N-x+1)\over\Gamma(N-x-n+1)}{}_2F_1(-n,-x; N-x-n+1; -q/p).\quad$	(7)

See also Orthogonal Polynomials

References

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 35-37, 1975.

Zelenkov, V. ``Krawtchouk Polynomial Home Page.'' http://www.isir.minsk.by/~zelenkov/physmath/kr_polyn/.