Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname ) are
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
associated with the
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
, introduced by .
The first few polynomials are (for ''q'' = 2):
:
:
:
:
The Kravchuk polynomials are a special case of the
Meixner polynomials
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol
In mathematics, th ...
of the first kind.
Definition
For any
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
''q'' and positive integer ''n'', define the Kravchuk polynomial
:
Properties
The Kravchuk polynomial has the following alternative expressions:
:
:
Symmetry relations
For integers
, we have that
:
Orthogonality relations
For non-negative integers ''r'', ''s'',
:
Generating function
The
generating series of Kravchuk polynomials is given as below. Here
is a formal variable.
:
See also
*
Krawtchouk matrix
*
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well a ...
References
*
*
*.
*.
*
External links
{{commons category
Krawtchouk Polynomials Home Pageat
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
Orthogonal polynomials