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Krawtchouk Polynomial

Let alpha(x) be a step function with the jump

j(x)=(N; x)p^xq^(N-x)
(1)

at x=0, 1, ..., N, where p>0,q>0, and p+q=1. Then the Krawtchouk polynomial is defined by

k_n^((p))(x,N)=sum_(nu=0)^(n)(-1)^(n-nu)(N-x; n-nu)(x; nu)p^(n-nu)q^nu,
(2)
=(-1)^n(N; n)p^n_2F_1(-n,-x;-N;1/p)
(3)
=((-1)^np^n)/(n!)(Gamma(N-x+1))/(Gamma(N-x-n+1))×_2F_1(-n,-x;N-x-n+1;(p-1)/p).
(4)

for n=0, 1, ..., N. The first few Krawtchouk polynomials are

k_0^((p))(x,N)=1
(5)
k_1^((p))(x,N)=-Np+x
(6)
k_2^((p))(x,N)=1/2[N^2p^2+x(2p+x-1)-Np(p+2x)].
(7)

Koekoek and Swarttouw (1998) define the Krawtchouk polynomial without the leading coefficient as

K_n(x;p,N)=_2F_1(-n,-x;-N;1/p).
(8)

The Krawtchouk polynomials have weighting function

w=(N!p^xq^(N-x))/(Gamma(1+x)Gamma(N+1-x)),
(9)

where Gamma(x) is the gamma function, recurrence relation

(n+1)k_(n+1)^((p))(x,N)+pq(N-n+1)k_(n-1)^((p))(x,N) =[x-n-(N-2)]k_n^((p))(x,N),
(10)

and squared norm

(N!)/(n!(N-n)!)(pq)^n.
(11)

It has the limit

lim_(N->infty)(2/(Npq))^(n/2)n!k_n^((p))(Np+sqrt(2Npq)s,N)=H_n(s),
(12)

where H_n(x) is a Hermite polynomial.

The Krawtchouk polynomials are a special case of the Meixner polynomials of the first kind.

See also

Hamming Scheme, Meixner Polynomial of the First Kind, Orthogonal Polynomials

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References

Koekoek, R. and Swarttouw, R. F. "Krawtchouk." §1.10 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 46-47, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998.Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.Schrijver, A. "A Comparison of the Delsarte and Lovász Bounds." IEEE Trans. Inform. Th. 25, 425-429, 1979.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 35-37, 1975.Zelenkov, V. "Krawtchouk Polynomials Home Page." http://www.geocities.com/orthpol/.

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Krawtchouk Polynomial

Cite this as:

Weisstein, Eric W. "Krawtchouk Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KrawtchoukPolynomial.html

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