Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к) are
discreteorthogonal polynomials associated with the
binomial distribution, introduced by
Mykhailo Kravchuk (
1929).
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 powerq 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
Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B. (1991), Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Berlin: Springer-Verlag,
ISBN3-540-51123-7,
MR1149380.
Levenshtein, Vladimir I. (1995), "Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", IEEE Transactions on Information Theory, 41 (5): 1303–1321,
doi:
10.1109/18.412678,
MR1366326.
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к) are
discreteorthogonal polynomials associated with the
binomial distribution, introduced by
Mykhailo Kravchuk (
1929).
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 powerq 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
Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B. (1991), Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Berlin: Springer-Verlag,
ISBN3-540-51123-7,
MR1149380.
Levenshtein, Vladimir I. (1995), "Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", IEEE Transactions on Information Theory, 41 (5): 1303–1321,
doi:
10.1109/18.412678,
MR1366326.