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Kravchuk polynomials
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<h2 id="definition">Definition</h2> <p>For any <a href="/facts/Prime_power/R0vo3uYp">prime power</a> <i>q</i> and positive integer <i>n</i>, define the Kravchuk polynomial K k ( x ; n , q ) = K k ( x ) = ∑ j = 0 k ( − 1 ) j ( q − 1 ) k − j ( x j ) ( n − x k − j ) = ∑ j = 0 k ( − 1 ) j ( q − 1 ) k − j x j _ j ! ( n − x ) k − j _ ( k − j ) ! {\displaystyle {\begin{aligned}{\mathcal {K}}_{k}(x;n,q)={\mathcal {K}}_{k}(x)={}&\sum _{j=0}^{k}(-1)^{j}(q-1)^{k-j}{\binom {x}{j}}{\binom {n-x}{k-j}}\\={}&\sum _{j=0}^{k}(-1)^{j}(q-1)^{k-j}{\frac {x^{\underline {j}}}{j!}}{\frac {(n-x)^{\underline {k-j}}}{(k-j)!}}\end{aligned}}} for k = 0 , 1 , … , n {\displaystyle k=0,1,\ldots ,n} . In the second line, the factors depending on x {\displaystyle x} have been rewritten in terms of <a href="/facts/Falling_factorial/sAxnmGoC">falling factorials</a>, to aid readers uncomfortable with non-integer arguments of binomial coefficients. </p> <h2 id="properties">Properties</h2> <p>The Kravchuk polynomial has the following alternative expressions: </p> K k ( x ; n , q ) = ∑ j = 0 k ( − q ) j ( q − 1 ) k − j ( n − j k − j ) ( x j ) . {\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-q)^{j}(q-1)^{k-j}{\binom {n-j}{k-j}}{\binom {x}{j}}.} K k ( x ; n , q ) = ∑ j = 0 k ( − 1 ) j q k − j ( n − k + j j ) ( n − x k − j ) . {\displaystyle {\mathcal {K}}_{k}(x;n,q)=\sum _{j=0}^{k}(-1)^{j}q^{k-j}{\binom {n-k+j}{j}}{\binom {n-x}{k-j}}.} <p>Note that there is more that merely recombination of material from the two binomial coefficients separating these from the above definition. In these formulae, only one term of the sum has degree k {\displaystyle k} , whereas in the definition all terms have degree k {\displaystyle k} . </p> <h3>Symmetry relations</h3> <p>For integers i , k ≥ 0 {\displaystyle i,k\geq 0} , we have that </p> ( q − 1 ) i ( n i ) K k ( i ; n , q ) = ( q − 1 ) k ( n k ) K i ( k ; n , q ) . {\displaystyle {\begin{aligned}(q-1)^{i}{n \choose i}{\mathcal {K}}_{k}(i;n,q)=(q-1)^{k}{n \choose k}{\mathcal {K}}_{i}(k;n,q).\end{aligned}}} <h3>Orthogonality relations</h3> <p>For non-negative integers <i>r</i>, <i>s</i>, </p> ∑ i = 0 n ( n i ) ( q − 1 ) i K r ( i ; n , q ) K s ( i ; n , q ) = q n ( q − 1 ) r ( n r ) δ r , s . {\displaystyle \sum _{i=0}^{n}{\binom {n}{i}}(q-1)^{i}{\mathcal {K}}_{r}(i;n,q){\mathcal {K}}_{s}(i;n,q)=q^{n}(q-1)^{r}{\binom {n}{r}}\delta _{r,s}.} <h3>Generating function</h3> <p>The <a href="/facts/Generating_series/K7UVjnjL">generating series</a> of Kravchuk polynomials is given as below. Here z {\displaystyle z} is a formal variable. </p> ( 1 + ( q − 1 ) z ) n − x ( 1 − z ) x = ∑ k = 0 ∞ K k ( x ; n , q ) z k . {\displaystyle {\begin{aligned}(1+(q-1)z)^{n-x}(1-z)^{x}&=\sum _{k=0}^{\infty }{\mathcal {K}}_{k}(x;n,q){z^{k}}.\end{aligned}}} <h3>Three term recurrence</h3> <p>The Kravchuk polynomials satisfy the three-term recurrence relation </p> x K k ( x ; n , q ) = − q ( n − k ) K k + 1 ( x ; n , q ) + ( q ( n − k ) + k ( 1 − q ) ) K k ( x ; n , q ) − k ( 1 − q ) K k − 1 ( x ; n , q ) . {\displaystyle {\begin{aligned}x{\mathcal {K}}_{k}(x;n,q)=-q(n-k){\mathcal {K}}_{k+1}(x;n,q)+(q(n-k)+k(1-q)){\mathcal {K}}_{k}(x;n,q)-k(1-q){\mathcal {K}}_{k-1}(x;n,q).\end{aligned}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Krawtchouk_matrix/evVnFlgJ">Krawtchouk matrix</a></li> <li><a href="/facts/Hermite_polynomials/MLQNY0ez">Hermite polynomials</a></li></ul> <ul><li><a href="/facts/Mikhail_Kravchuk/a3zp9oKS">Kravchuk, M.</a> (1929), <a href="http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN">"Sur une généralisation des polynomes d'Hermite."</a>, <i>Comptes Rendus Mathématique</i> (in French), 189: 620–622, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:55.0799.01">55.0799.01</a></li> <li>Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), <a href="http://dlmf.nist.gov/18.19">"Hahn Class: Definitions"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li> <li>Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B. (1991), <i>Classical Orthogonal Polynomials of a Discrete Variable</i>, Springer Series in Computational Physics, Berlin: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-51123-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1149380">1149380</a>.</li> <li><a href="/facts/Vladimir_Levenshtein/hGbzUR8e">Levenshtein, Vladimir I.</a> (1995), "Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces", <i>IEEE Transactions on Information Theory</i>, 41 (5): 1303–1321, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F18.412678">10.1109/18.412678</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1366326">1366326</a>.</li> <li>MacWilliams, F. J.; Sloane, N. J. A. (1977), <a href="https://archive.org/details/theoryoferrorcor0000macw"><i>The Theory of Error-Correcting Codes</i></a>, North-Holland, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-85193-3</li></ul> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Kravchuk polynomials. <ul><li><a href="https://web.archive.org/web/20070205055023/http://orthpol.narod.ru/">Krawtchouk Polynomials Home Page</a></li> <li><a href="http://mathworld.wolfram.com/KrawtchoukPolynomial.html">"Krawtchouk polynomial"</a> at <a href="/facts/MathWorld/yc1jodbq">MathWorld</a></li></ul>