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:
![{\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]}](../I/m/b11373862068ca4c8c558e6b941a8c38d137003f.svg)
![{\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].}](../I/m/3fe08d1f2b18ef9c0cfe249575de9c7fd6532df1.svg)
In general, for positive integer , the entries are given via the generating function
where the row and column indices and run from to .
These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, .[3]
See also
References
- ↑ Bose, N. (1985). Digital Filters: Theory and Applications. New York: North-Holland Elsevier. ISBN 0-444-00980-9.
- ↑ Feinsilver, P.; Kocik, J. (2004). Krawtchouk polynomials and Krawtchouk matrices. Recent Advances in Applied Probability. Springer-Verlag. arXiv:quant-ph/0702073. Bibcode:2007quant.ph..2073F.
- ↑ "Hahn Class: Definitions". Digital Library of Mathematical Functions.
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