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):
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
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.
Three term recurrence
The Kravchuk polynomials satisfy the three-term recurrence relation
See also
Krawtchouk matrix
Hermite polynomials
References
. .