## Krawtchouk Polynomial

Let be a Step Function with the Jump

 (1)

at , 1, ..., , where , and . Then

 (2)

for , 1, ..., . It has Weight Function
 (3)

where is the Gamma Function, Recurrence Relation

 (4)

and squared norm
 (5)

It has the limit
 (6)

where is a Hermite Polynomial, and is related to the Hypergeometric Function by

 (7)

See also Orthogonal Polynomials

References

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 35-37, 1975.

Zelenkov, V. Krawtchouk Polynomial Home Page.'' http://www.isir.minsk.by/~zelenkov/physmath/kr_polyn/.

© 1996-9 Eric W. Weisstein
1999-05-26