Krawtchouk matrices
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 ]](/2012-wikipedia_en_all_nopic_01_2012/I/4a6dec20e3877021e0cb2803eeb50418.png)
![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 ].](/2012-wikipedia_en_all_nopic_01_2012/I/ebf47d91d069f537dcdaf6240a5a8579.png)
See also
References
- ^ N. Bose, “Digital Filters: Theory and Applications” [North-Holland Elsevier, N.Y., 1985]
- ^ P. Feinsilver, J. Kocik: Krawtchouk polynomials and Krawtchouk matrices, Recent advances in applied probability, Springer-Verlag, October, 2004