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Kravchuk polynomials : ウィキペディア英語版
Kravchuk polynomials
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian name "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by .
The first few polynomials are (for ''q''=2):
* \mathcal_0(x; n) = 1
* \mathcal_1(x; n) = -2x + n
* \mathcal_2(x; n) = 2x^2 - 2nx +
* \mathcal_3(x; n) = -\fracx^3 + 2nx^2 - (n^2 - n + \frac)x + .
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
==Definition==
For any prime power ''q'' and positive integer ''n'', define the Kravchuk polynomial
:\mathcal_k(x; n) = \mathcal_k(x) = \sum_^(-1)^j (q-1)^ \binom \binom, \quad k=0,1, \ldots, n.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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