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Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к discrete orthogonal polynomials associated with the binomial distribution , introduced by Mykhailo Kravchuk (1929 ).
The first few polynomials are (for q = 2):
K
0
(
x
;
n
)
=
1
,
{\displaystyle {\mathcal {K}}_{0}(x;n)=1,}
K
1
(
x
;
n
)
=
−
2
x
+
n
,
{\displaystyle {\mathcal {K}}_{1}(x;n)=-2x+n,}
K
2
(
x
;
n
)
=
2
x
2
−
2
n
x
+
(
n
2
)
,
{\displaystyle {\mathcal {K}}_{2}(x;n)=2x^{2}-2nx+{\binom {n}{2}},}
K
3
(
x
;
n
)
=
−
4
3
x
3
+
2
n
x
2
−
(
n
2
−
n
+
2
3
)
x
+
(
n
3
)
.
{\displaystyle {\mathcal {K}}_{3}(x;n)=-{\frac {4}{3}}x^{3}+2nx^{2}-(n^{2}-n+{\frac {2}{3}})x+{\binom {n}{3}}.}
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
For any prime power q and positive integer n , define the Kravchuk polynomial
K
k
(
x
;
n
,
q
)
=
K
k
(
x
)
=
∑
j
=
0
k
(
−
1
)
j
(
q
−
1
)
k
−
j
(
x
j
)
(
n
−
x
k
−
j
)
,
k
=
0
,
1
,
…
,
n
.
{\displaystyle {\mathcal {K}}_{k}(x;n,q)={\mathcal {K}}_{k}(x)=\sum _{j=0}^{k}(-1)^{j}(q-1)^{k-j}{\binom {x}{j}}{\binom {n-x}{k-j}},\quad k=0,1,\ldots ,n.}
The Kravchuk polynomial has the following alternative expressions:
K
k
(
x
;
n
,
q
)
=
∑
j
=
0
k
(
−
q
)
j
(
q
−
1
)
k
−
j
(
n
−
j
k
−
j
)
(
x
j
)
.
{\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-q)^{j}(q-1)^{k-j}{\binom {n-j}{k-j}}{\binom {x}{j}}.}
K
k
(
x
;
n
,
q
)
=
∑
j
=
0
k
(
−
1
)
j
q
k
−
j
(
n
−
k
+
j
j
)
(
n
−
x
k
−
j
)
.
{\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-1)^{j}q^{k-j}{\binom {n-k+j}{j}}{\binom {n-x}{k-j}}.}
For integers
i
,
k
≥
0
{\displaystyle i,k\geq 0}
(
q
−
1
)
i
(
n
i
)
K
k
(
i
;
n
,
q
)
=
(
q
−
1
)
k
(
n
k
)
K
i
(
k
;
n
,
q
)
.
{\displaystyle {\begin{aligned}(q-1)^{i}{n \choose i}{\mathcal {K}}_{k}(i;n,q)=(q-1)^{k}{n \choose k}{\mathcal {K}}_{i}(k;n,q).\end{aligned}}}
Orthogonality relations [ edit ] For non-negative integers r , s ,
∑
i
=
0
n
(
n
i
)
(
q
−
1
)
i
K
r
(
i
;
n
,
q
)
K
s
(
i
;
n
,
q
)
=
q
n
(
q
−
1
)
r
(
n
r
)
δ
r
,
s
.
{\displaystyle \sum _{i=0}^{n}{\binom {n}{i}}(q-1)^{i}{\mathcal {K}}_{r}(i;n,q){\mathcal {K}}_{s}(i;n,q)=q^{n}(q-1)^{r}{\binom {n}{r}}\delta _{r,s}.}
Generating function [ edit ] The generating series of Kravchuk polynomials is given as below. Here
z
{\displaystyle z}
(
1
+
(
q
−
1
)
z
)
n
−
x
(
1
−
z
)
x
=
∑
k
=
0
∞
K
k
(
x
;
n
,
q
)
z
k
.
{\displaystyle {\begin{aligned}(1+(q-1)z)^{n-x}(1-z)^{x}&=\sum _{k=0}^{\infty }{\mathcal {K}}_{k}(x;n,q){z^{k}}.\end{aligned}}}
Three term recurrence [ edit ] The Kravchuk polynomials satisfy the three-term recurrence relation
x
K
k
(
x
;
n
,
q
)
=
−
q
(
n
−
k
)
K
k
+
1
(
x
;
n
,
q
)
+
(
q
(
n
−
k
)
+
k
(
1
−
q
)
)
K
k
(
x
;
n
,
q
)
−
k
(
1
−
q
)
K
k
−
1
(
x
;
n
,
q
)
.
{\displaystyle {\begin{aligned}x{\mathcal {K}}_{k}(x;n,q)=-q(n-k){\mathcal {K}}_{k+1}(x;n,q)+(q(n-k)+k(1-q)){\mathcal {K}}_{k}(x;n,q)-k(1-q){\mathcal {K}}_{k-1}(x;n,q).\end{aligned}}}
Kravchuk, M. (1929), "Sur une généralisation des polynomes d'Hermite." , Comptes Rendus Mathématique (in French), 189 : 620–622, JFM 55.0799.01 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B. (1991), Classical Orthogonal Polynomials of a Discrete Variable , Springer Series in Computational Physics, Berlin: Springer-Verlag, ISBN 3-540-51123-7 MR 1149380 Levenshtein, Vladimir I. (1995), "Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", IEEE Transactions on Information Theory , 41 (5): 1303–1321, doi :10.1109/18.412678 , MR 1366326 MacWilliams, F. J.; Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes ISBN 0-444-85193-3 
