Krawtchouk Polynomial

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

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

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,
=(-1)^n(N; n)p^n_2F_1(-n,-x;-N;1/p)

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


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


The Krawtchouk polynomials have weighting function


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


and squared norm


It has the limit


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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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."

Referenced on Wolfram|Alpha

Krawtchouk Polynomial

Cite this as:

Weisstein, Eric W. "Krawtchouk Polynomial." From MathWorld--A Wolfram Web Resource.

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