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# Krawtchouk Polynomial

Let be a step function with the jump

 (1)

at , 1, ..., , where , and . Then the Krawtchouk polynomial is defined by

 (2) (3) (4)

for , 1, ..., . The first few Krawtchouk polynomials are

 (5) (6) (7)

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

 (8)

The Krawtchouk polynomials have weighting function

 (9)

where is the gamma function, recurrence relation

 (10)

and squared norm

 (11)

It has the limit

 (12)

where is a Hermite polynomial.

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

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 -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/.

## Referenced on Wolfram|Alpha

Krawtchouk Polynomial

## Cite this as:

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