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Krawtchouk polynomials

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Polynomials orthogonal on the finite system of $N+1$ integer points whose distribution function $\sigma(z)$ is a step function with discontinuities: $$ \sigma(x+) - \sigma(x-) = \binom{N}{x} p^x q^{N-x} \,,\ \ \ x=0,\ldots,N $$

where $\binom{\cdot}{\cdot}$ is the binomial coefficient, $p,q > 0$ and $p+q = 1$. The Krawtchouk polynomials are given by the formulas $$ P_n(x) = \left[ \binom{N}{x} \right]^{-1/2} (pq)^{-n/2} \sum_{k=0}^n (-1)^{n-k} \binom{N-x}{n-k} \binom{x}{k} p^{n-k} q^k \ . $$

The concept is due to M.F. Krawtchouk [1].

References

[1] M.F. Krawtchouk, "Sur une généralisation des polynômes d'Hermite" C.R. Acad. Sci. Paris , 189 (1929) pp. 620–622
[2] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)


Comments

Krawtchouk polynomials can be written as hypergeometric functions of type ${}_2F_1$. The unitarity relations for the matrix elements of the irreducible unitary representations of the group $SU(2)$ can be rewritten as the orthogonality relations for the Krawtchouk polynomials, cf. [a2], [a3]. These polynomials have also an interpretation as spherical functions on wreath products of symmetric groups, cf. [a4], where $q$-Krawtchouk polynomials are also treated. Coding theorists rather (but equivalently) relate them to Hamming schemes, where Krawtchouk polynomials are used for dealing with problems about perfect codes, cf. [a1].

References

[a1] J.H. van Lint, "Introduction to coding theory" , Springer (1982)
[a2] T.H. Koornwinder, "Krawtchouk polynomials, a unification of two different group theoretic interpretations" SIAM J. Math. Anal. , 13 (1982) pp. 1011–1023
[a3] V.B. Uvarov, "Special functions of mathematical physics" , Birkhäuser (1988) (Translated from Russian)
[a4] D. Stanton, "Orthogonal polynomials and Chevalley groups" R.A. Askey (ed.) T.H. Koornwinder (ed.) W. Schempp (ed.) , Special functions: group theoretical aspects and applications , Reidel (1984) pp. 87–128
How to Cite This Entry:
Krawtchouk polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Krawtchouk_polynomials&oldid=14107
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article