Krawtchouk matrices

Krawtchouk matrices

Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. [ N. Bose, “Digital Filters: Theory and Applications” [North-Holland Elsevier, N.Y., 1985] ] [ [http://aps.arxiv.org/abs/quant-ph/0702073 P. Feinsilver, J. Kocik: Krawtchouk polynomials and Krawtchouk matrices, "Recent advances in applied probability", Springer-Verlag, October, 2004] ] The Krawtchouk matrix "K(n)" is an "(n+1)×(n+1)" matrix. Here are the first few examples:


K^{(0)}=egin{bmatrix} 1 end{bmatrix}qquadK^{(1)}=left [ egin{array}{rr} 1&1\ 1&-1end{array} ight ] qquadK^{(2)}=left [ egin{array}{rrr} 1&1&1\ 2&0&-2\ 1&-1&1end{array} ight ] qquadK^{(3)}=left [ egin{array}{rrrr} 1&1&1&1\ 3&1&-1&-3\ 3&-1&-1&3\ 1&-1&1&-1end{array} ight ]

K^{(4)}=left [ egin{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&1end{array} ight ] qquadK^{(5)}=left [ egin{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&-1end{array} ight ]

ee also

*

References

External links

* [http://chanoir.math.siu.edu/wiki/published Krawtchouk encyclopedia]


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