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 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
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