Let
be a step function with the jump

(1)

at ,
1, ..., ,
where ,
and .
Then the Krawtchouk polynomial is defined by
for ,
1, ..., .
The first few Krawtchouk polynomials are
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.
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 AskeyScheme of Hypergeometric Orthogonal Polynomials and its
Analogue.
Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics
and Informatics Report 9817, pp. 4647, 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: SpringerVerlag,
1992.Schrijver, A. "A Comparison of the Delsarte and Lovász
Bounds." IEEE Trans. Inform. Th. 25, 425429, 1979.Szegö,
G. Orthogonal
Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 3537, 1975.Zelenkov,
V. "Krawtchouk Polynomials Home Page." http://www.geocities.com/orthpol/.Referenced
on WolframAlpha
Krawtchouk Polynomial
Cite this as:
Weisstein, Eric W. "Krawtchouk Polynomial."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/KrawtchoukPolynomial.html
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