Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname ) are discrete orthogonal polynomials associated with the binomial distribution, introduced by .
The first few polynomials are (for ''q'' = 2):
: $\mathcal_0(x; n) = 1,$
: $\mathcal_1(x; n) = -2x + n,$
: $\mathcal_2(x; n) = 2x^2 - 2nx + \binom,$
: $\mathcal_3(x; n) = -\fracx^3 + 2nx^2 - (n^2 - n + \frac)x + \binom.$

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

:$\mathcal_k(x; n,q) = \mathcal_k(x) = \sum_^(-1)^j (q-1)^ \binom \binom, \quad k=0,1, \ldots, n.$

Properties

The Kravchuk polynomial has the following alternative expressions:

:$\mathcal_k(x; n,q) = \sum_^(-q)^j (q-1)^ \binom \binom.$
:$\mathcal_k(x; n,q) = \sum_^(-1)^j q^ \binom \binom.$

Symmetry relations

For integers $i,k \ge 0$, we have that
:$\begin (q-1)^ \mathcal_k(i;n,q) = (q-1)^ \mathcal_i(k;n,q). \end$

Orthogonality relations

For non-negative integers ''r'', ''s'',

:$\sum_^n\binom(q-1)^i\mathcal_r(i; n,q)\mathcal_s(i; n,q) = q^n(q-1)^r\binom\delta_.$

Generating function

The generating series of Kravchuk polynomials is given as below. Here $z$ is a formal variable.
:$\begin (1+(q-1)z)^(1-z)^x &= \sum_^\infty \mathcal_k(x;n,q) . \end$

See also

* Krawtchouk matrix
* Hermite polynomials

References

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External links

{{commons category
Krawtchouk Polynomials Home Page
at MathWorld

Orthogonal polynomials