Superincreasing sequence

In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K^(N) is an (N+1)×(N+1) matrix. Here are the first few examples:

$K^{(0)}=\left[\begin{array}{c}1\end{array}\right]K^{(1)}=\left[\begin{array}{rr}1& 1\\ 1& -1\end{array}\right]K^{(2)}=\left[\begin{array}{rrr}1& 1& 1\\ 2& 0& -2\\ 1& -1& 1\end{array}\right]K^{(3)}=\left[\begin{array}{rrrr}1& 1& 1& 1\\ 3& 1& -1& -3\\ 3& -1& -1& 3\\ 1& -1& 1& -1\end{array}\right]$

$K^{(4)}=\left[\begin{array}{rrrrr}1& 1& 1& 1& 1\\ 4& 2& 0& -2& -4\\ 6& 0& -2& 0& 6\\ 4& -2& 0& 2& -4\\ 1& -1& 1& -1& 1\end{array}\right]K^{(5)}=\left[\begin{array}{rrrrrr}1& 1& 1& 1& 1& 1\\ 5& 3& 1& -1& -3& -5\\ 10& 2& -2& -2& 2& 10\\ 10& -2& -2& 2& 2& -10\\ 5& -3& 1& 1& -3& 5\\ 1& -1& 1& -1& 1& -1\end{array}\right].$

In general, for positive integer $N$, the entries $K_{ij}^{(N)}$ are given via the generating function

$(1+v{)}^{N-j}(1-v{)}^j=\sum _i{v}^i{K}_{ij}^{(N)}$

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, $p=1/2$.

See also

• Square matrix

References

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External links

• Krawtchouk encyclopedia

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