Krawtchouk Polynomial
المؤلف:
Koekoek, R. and Swarttouw, R. F.
المصدر:
"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
الجزء والصفحة:
...
4-8-2019
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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
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
![(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),](http://mathworld.wolfram.com/images/equations/KrawtchoukPolynomial/NumberedEquation4.gif) |
(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.
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/.
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