# Krawtchouk matrices

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In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points.[1] [2] The Krawtchouk matrix K(N) is an (N+1)×(N+1) matrix. Here are the first few examples:

${\displaystyle K^{(0)}={\begin{bmatrix}1\end{bmatrix}}\qquad K^{(1)}=\left[{\begin{array}{rr}1&1\\1&-1\end{array}}\right]\qquad K^{(2)}=\left[{\begin{array}{rrr}1&1&1\\2&0&-2\\1&-1&1\end{array}}\right]\qquad K^{(3)}=\left[{\begin{array}{rrrr}1&1&1&1\\3&1&-1&-3\\3&-1&-1&3\\1&-1&1&-1\end{array}}\right]}$

${\displaystyle K^{(4)}=\left[{\begin{array}{rrrrr}1&1&1&1&1\\4&2&0&-2&-4\\6&0&-2&0&6\\4&-2&0&2&-4\\1&-1&1&-1&1\end{array}}\right]\qquad K^{(5)}=\left[{\begin{array}{rrrrrr}1&1&1&1&1&1\\5&3&1&-1&-3&-5\\10&2&-2&-2&2&10\\10&-2&-2&2&2&-10\\5&-3&1&1&-3&5\\1&-1&1&-1&1&-1\end{array}}\right].}$

In general, for positive integer ${\displaystyle N}$, the entries ${\displaystyle K_{ij}^{(N)}}$ are given via the generating function

${\displaystyle (1+v)^{N-j}\,(1-v)^{j}=\sum _{i}v^{i}K_{ij}^{(N)}}$

where the row and column indices ${\displaystyle i}$ and ${\displaystyle j}$ run from ${\displaystyle 0}$ to ${\displaystyle N}$.

These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, ${\displaystyle p=1/2}$.[3]