Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Ukrainian: Кравчу́к) are discrete orthogonal polynomials associated with the binomial distribution, introduced by .The first few polynomials are (for q = 2):
l{K}0(x;n)=1,
l{K}1(x;n)=-2x+n,
l{K}2(x;n)=2x2-2nx+\binom{n}{2},
l{K}3(x;n)=-
| 4 | |
| 3 |
x3+2nx2-(n2-n+
| 2 | |
| 3 |
)x+\binom{n}{3}.
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
For any prime power q and positive integer n, define the Kravchuk polynomial
l{K}k(x;n,q)=l{K}k(x)=
| k | |
| \sum | |
| j=0 |
(-1)j(q-1)k-j\binom{x}{j}\binom{n-x}{k-j}, k=0,1,\ldots,n.
The Kravchuk polynomial has the following alternative expressions:
l{K}k(x;n,q)=
| k | |
| \sum | |
| j=0 |
(-q)j(q-1)k-j\binom{n-j}{k-j}\binom{x}{j}.
l{K}k(x;n,q)=
| k | |
| \sum | |
| j=0 |
(-1)jqk-j\binom{n-k+j}{j}\binom{n-x}{k-j}.
For integers
i,k\ge0
\begin{align} (q-1)i{n\choosei}l{K}k(i;n,q)=(q-1)k{n\choosek}l{K}i(k;n,q). \end{align}
For non-negative integers r, s,
| n\binom{n}{i}(q-1) | |
| \sum | |
| i=0 |
| il{K} | |
| r(i; |
n,q)l{K}s(i;n,q)=qn(q-1)
| r\binom{n}{r}\delta | |
| r,s |
.
The generating series of Kravchuk polynomials is given as below. Here
z
\begin{align} (1+(q-1)z)n-x(1-z)x&=
| infty | |
| \sum | |
| k=0 |
l{K}k(x;n,q){zk}. \end{align}
The Kravchuk polynomials satisfy the three-term recurrence relation
\begin{align} xl{K}k(x;n,q)=-q(n-k)l{K}k+1(x;n,q)+(q(n-k)+k(1-q))l{K}k(x;n,q)-k(1-q)l{K}k-1(x;n,q). \end{align}