Kravchuk polynomials


Kravchuk polynomials or Krawtchouk polynomials are discrete orthogonal polynomials associated with the binomial distribution, introduced by .
The first few polynomials are :
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.