Krawtchouk matrices について

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Krawtchouk matrices
In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points.〔N. Bose, “Digital Filters: Theory and Applications” (Elsevier, N.Y., 1985 )〕
〔(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^=\begin                     1               \end\qquadK^=\left (\begin                  1&1\\                  1&-1\end\right) \qquadK^=\left (\begin                  1&1&1\\                  2&0&-2\\                  1&-1&1\end\right) \qquadK^=\left (\begin               1&1&1&1\\               3&1&-1&-3\\               3&-1&-1&3\\               1&-1&1&-1\end\right)

K^=\left (\begin              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\right) \qquadK^=\left (\begin                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\right).

In general, for positive integer

N

, the entries

K^_

are given via the generating function
:

(1+v)^\,(1-v)^j=\sum_i v^i K^_

where the row and column indices

i

and

j

run from

0

N

.
These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions,

p=1/2

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