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| '''Kravchuk polynomials''' or '''Krawtchouk polynomials''' (also written using several other transliterations of the Ukrainian name "Кравчу́к") are [[discrete orthogonal polynomials|discrete]] [[orthogonal polynomials]] associated with the [[binomial distribution]], introduced by {{harvs|txt|authorlink=Mikhail Kravchuk|first=Mikhail|last=Kravchuk|year= 1929}}.
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| The first few polynomials are: | |
| * <math>\mathcal{K}_0(x; n) = 1</math>
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| * <math>\mathcal{K}_1(x; n) = -2x + n</math>
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| * <math>\mathcal{K}_2(x; n) = 2x^2 - 2nx + {n\choose 2}</math>
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| * <math>\mathcal{K}_3(x; n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + {n \choose 3}.</math>
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| The Kravchuk polynomials are a special case of the [[Meixner polynomials]] of the first kind.
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| ==Definition==
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| For any [[prime power]] ''q'' and positive integer ''n'', define the Kravchuk polynomial
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| :<math>\mathcal{K}_k(x; n) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n.</math>
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| ==Properties==
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| The Kravchuk polynomial has following alternative expressions:
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| :<math>\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}. </math>
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| :<math>\mathcal{K}_k(x; n) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}. </math>
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| ===Orthogonality relations===
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| For nonnegative integers ''r'', ''s'',
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| :<math>\sum_{i=0}^n\binom{n}{j}(q-1)^i\mathcal{K}_r(i; n)\mathcal{K}_s(i; n) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}. </math> | |
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| ==See also==
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| * [[Hermite polynomials]]
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| ==References==
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| <references/>
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| *{{Citation | last1=Kravchuk | first1=M. | | authorlink = Mikhail Kravchuk | title=Sur une généralisation des polynomes d'Hermite. | url=http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN | language=French | jfm=55.0799.01 | year=1929 | journal=Comptes Rendus Mathematique | volume=189 | pages=620–622}}
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| *{{dlmf|id=18.19|title=Hahn Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| *{{citation
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| | last1 = Nikiforov | first1 = A. F.
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| | last2 = Suslov | first2 = S. K.
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| | last3 = Uvarov | first3 = V. B.
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| | isbn = 3-540-51123-7
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| | location = Berlin
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| | mr = 1149380
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| | publisher = Springer-Verlag
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| | series = Springer Series in Computational Physics
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| | title = Classical Orthogonal Polynomials of a Discrete Variable
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| | year = 1991}}.
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| *{{citation
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| | last = Levenshtein | first = Vladimir I. | author-link = Vladimir Levenshtein
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| | doi = 10.1109/18.412678
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| | issue = 5
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| | journal = IEEE Transactions on Information Theory
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| | mr = 1366326
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| | pages = 1303–1321
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| | title = Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces
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| | volume = 41
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| | year = 1995}}.
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| *{{Citation | author=F. J. MacWilliams | coauthors=N. J. A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3}}
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| ==External links==
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| *[http://orthpol.narod.ru/ Krawtchouk Polynomials Home Page]
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| *[http://mathworld.wolfram.com/KrawtchoukPolynomial.html "Krawtchouk polynomial"] at [[MathWorld]]
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| [[Category:Orthogonal polynomials]]
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The writer is called Araceli Gulledge. Interviewing is how I make a living and it's something I really appreciate. Delaware has always been my living place and will by no means transfer. Climbing is what adore doing.
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