克拉夫丘克多項式 以超幾何函數 定義如下
Kravchuk polynomials animation Kravchuk polynomials animation
K
n
(
x
;
p
,
N
)
=
(
N
n
)
−
1
∗
{\displaystyle K_{n}(x;p,N)={N \choose n}^{-1}*}
∑
k
=
0
n
(
N
−
x
n
−
k
)
∗
(
x
k
)
∗
(
1
−
1
/
p
)
k
{\displaystyle \sum _{k=0}^{n}{N-x \choose n-k}*{x \choose k}*(1-1/p)^{k}}
2
F
1
(
−
n
,
−
x
;
N
;
1
/
p
)
{\displaystyle _{2}F_{1}(-n,-x;N;1/p)}
K
0
(
x
,
p
,
N
)
=
1
{\displaystyle K_{0}(x,p,N)=1}
K
1
(
x
,
p
,
N
)
=
1
−
x
p
∗
N
{\displaystyle K_{1}(x,p,N)=1-{\frac {x}{p*N}}}
K
2
(
x
,
p
,
N
)
=
1
+
(
−
2
p
∗
N
+
1
(
p
2
∗
N
∗
(
−
N
+
1
)
)
∗
x
−
x
2
(
p
2
∗
N
∗
(
−
N
+
1
)
)
{\displaystyle K_{2}(x,p,N)=1+({\frac {-2}{p*N}}+{\frac {1}{(p^{2}*N*(-N+1)}})*x-{\frac {x^{2}}{(p^{2}*N*(-N+1))}}}
K
3
(
x
,
p
,
N
)
=
1
+
(
−
3
(
p
∗
N
)
+
3
(
p
2
∗
N
∗
(
−
N
+
1
)
)
−
2
(
p
3
∗
N
∗
(
−
N
+
1
)
∗
(
−
N
+
2
)
)
)
∗
x
+
(
−
3
/
(
p
2
∗
N
∗
(
−
N
+
1
)
)
+
3
/
(
p
3
∗
N
∗
(
−
N
+
1
)
∗
(
−
N
+
2
)
)
)
∗
x
2
−
x
3
/
(
p
3
∗
N
∗
(
−
N
+
1
)
∗
(
−
N
+
2
)
)
{\displaystyle K_{3}(x,p,N)=1+({\frac {-3}{(p*N)}}+{\frac {3}{(p^{2}*N*(-N+1))}}-{\frac {2}{(p^{3}*N*(-N+1)*(-N+2))}})*x+(-3/(p^{2}*N*(-N+1))+3/(p^{3}*N*(-N+1)*(-N+2)))*x^{2}-x^{3}/(p^{3}*N*(-N+1)*(-N+2))}
量子Q克拉夫丘克多項式 → 克拉夫丘克多項式 ：
lim
a
→
1
=
K
n
q
t
m
(
q
−
x
;
p
,
N
;
q
)
=
K
n
(
x
;
p
−
1
,
N
)
{\displaystyle \lim _{a\to 1}=K_{n}^{qtm}(q^{-}{x};p,N;q)=K_{n}(x;p^{-1},N)}
令雙Q克拉夫丘克多項式
c
=
1
=
p
−
1
{\displaystyle c=1=p^{-1}}
克拉夫丘克多項式
Kravchuk, M. , Sur une généralisation des polynomes d'Hermite. , Comptes Rendus Mathematique, 1929, 189 : 620–622, JFM 55.0799.01 （法語） Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Hahn Class: Definitions , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Berlin: Springer-Verlag, 1991, ISBN 3-540-51123-7MR 1149380 Levenshtein, Vladimir I. , Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Transactions on Information Theory, 1995, 41 (5): 1303–1321, MR 1366326 doi:10.1109/18.412678 F. J. MacWilliams; N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977, ISBN 0-444-85193-3 
