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:



 
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 ]



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 N, the entries K^{(N)}_{ij} are given via the generating function

 (1+v)^{N-j}\,(1-v)^j=\sum_i v^i K^{(N)}_{ij}

where the row and column indices i and j run from 0 to N.

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

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