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
)
=
[
1
]
K
(
1
)
=
[
1
1
1
−
1
]
K
(
2
)
=
[
1
1
1
2
0
−
2
1
−
1
1
]
K
(
3
)
=
[
1
1
1
1
3
1
−
1
−
3
3
−
1
−
1
3
1
−
1
1
−
1
]
K
(
4
)
=
[
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
]
K
(
5
)
=
[
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
]
.
K
(
0
)
=
[
1
]
K
(
1
)
=
[
1
1
1
−
1
]
K
(
2
)
=
[
1
1
1
2
0
−
2
1
−
1
1
]
K
(
3
)
=
[
1
1
1
1
3
1
−
1
−
3
3
−
1
−
1
3
1
−
1
1
−
1
]
K
(
4
)
=
[
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
]
K
(
5
)
=
[
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
]
.
In general, for positive integer
N
, the entries
K
i
j
(
N
)
are given via the generating function
(
1
+
v
)
N
−
j
(
1
−
v
)
j
=
∑
i
v
i
K
i
j
(
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
.
