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 have been rewritten in terms of falling factorials, to aid readers uncomfortable with non-integer arguments of binomial coefficients.

Properties

The Kravchuk polynomial has the following alternative expressions:

.

Symmetry relations

, we have that

Orthogonality relations

For non-negative integers r, s,

Generating function

The generating series is a formal variable.

Three term recurrence

The Kravchuk polynomials satisfy the three-term recurrence relation

See also

• Krawtchouk matrix

• Hermite polynomials

References

• .

• .

External links

• Krawtchouk Polynomials Home Page

• "Krawtchouk polynomial" at MathWorld