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Difference between revisions of "Krawtchouk polynomials"

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Polynomials orthogonal on the finite system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558301.png" /> integer points whose distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558302.png" /> is a step function with discontinuities:
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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:
 +
$$
 +
\sigma(x+) - \sigma(x-) = \binom{N}{x} p^x q^{N-x} \,,\ \ \ x=0,\ldots,N
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558303.png" /></td> </tr></table>
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where $\binom{\cdot}{\cdot}$ is the [[Binomial coefficients|binomial coefficient]], $p,q > 0$ and $p+q = 1$. The Krawtchouk polynomials are given by the formulas
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558304.png" /> is the binomial coefficient (cf. [[Binomial coefficients|Binomial coefficients]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558306.png" />. 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 \ .
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558307.png" /></td> </tr></table>
 
  
 
The concept is due to M.F. Krawtchouk [[#References|[1]]].
 
The concept is due to M.F. Krawtchouk [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Krawtchouk,  "Sur une généralisation des polynômes d'Hermite"  ''C.R. Acad. Sci. Paris'' , '''189'''  (1929)  pp. 620–622</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Krawtchouk,  "Sur une généralisation des polynômes d'Hermite"  ''C.R. Acad. Sci. Paris'' , '''189'''  (1929)  pp. 620–622</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
Krawtchouk polynomials can be written as hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]) of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558308.png" />. The unitarity relations for the matrix elements of the irreducible unitary representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k0558309.png" /> can be rewritten as the orthogonality relations for the Krawtchouk polynomials, cf. [[#References|[a2]]], [[#References|[a3]]]. These polynomials have also an interpretation as spherical functions on wreath products (cf. [[Wreath product|Wreath product]]) of symmetric groups, cf. [[#References|[a4]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055830/k05583010.png" />-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. [[#References|[a1]]].
+
Krawtchouk polynomials can be written as [[hypergeometric function]]s 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. [[#References|[a2]]], [[#References|[a3]]]. These polynomials have also an interpretation as spherical functions on [[wreath product]]s of symmetric groups, cf. [[#References|[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. [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. van Lint,  "Introduction to coding theory" , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Koornwinder,  "Krawtchouk polynomials, a unification of two different group theoretic interpretations"  ''SIAM J. Math. Anal.'' , '''13'''  (1982)  pp. 1011–1023</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.B. Uvarov,  "Special functions of mathematical physics" , Birkhäuser  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. van Lint,  "Introduction to coding theory" , Springer  (1982)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Koornwinder,  "Krawtchouk polynomials, a unification of two different group theoretic interpretations"  ''SIAM J. Math. Anal.'' , '''13'''  (1982)  pp. 1011–1023</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.B. Uvarov,  "Special functions of mathematical physics" , Birkhäuser  (1988)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 20:05, 4 February 2016

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=37654
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article