Kravchuk polynomials

For any prime power q and positive integer n, define the Kravchuk polynomial

\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 $k=0,1,\; \backslash ldots,\; n$. In the second line, the factors depending on $x$ have been rewritten in terms of falling factorials, to aid readers uncomfortable with non-integer arguments of binomial coefficients.

The Kravchuk polynomial has the following alternative expressions:

:$\backslash mathcal\{K\}\_k(x;\; n,q)\; =\; \backslash sum\_\{j=0\}^\{k\}(-q)^j\; (q-1)^\{k-j\}\; \backslash binom\; \{n-j\}\{k-j\}\; \backslash binom\{x\}\{j\}.$

:$\backslash mathcal\{K\}\_k(x;\; n,q)\; =\; \backslash sum\_\{j=0\}^\{k\}(-1)^j\; q^\{k-j\}\; \backslash binom\; \{n-k+j\}\{j\}\; \backslash binom\{n-x\}\{k-j\}.$

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 $k$, whereas in the definition all terms have degree $k$.

For integers $i,k\; \backslash ge\; 0$, we have that

:$\backslash begin\{align\}$

(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{align}

For non-negative integers r, s,

:$\backslash sum\_\{i=0\}^n\backslash binom\{n\}\{i\}(q-1)^i\backslash mathcal\{K\}\_r(i;\; n,q)\backslash mathcal\{K\}\_s(i;\; n,q)\; =\; q^n(q-1)^r\backslash binom\{n\}\{r\}\backslash delta\_\{r,s\}.$

The generating series of Kravchuk polynomials is given as below. Here $z$ is a formal variable.

:$\backslash begin\{align\}$

(1+(q-1)z)^{n-x}(1-z)^x &= \sum_{k=0}^\infty \mathcal{K}_k(x;n,q) {z^k}.

\end{align}

The Kravchuk polynomials satisfy the three-term recurrence relation

:$\backslash begin\{align\}$

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{align}

• Krawtchouk matrix
• Hermite polynomials

• {{Citation | last1=Kravchuk | first1=M. | authorlink = Mikhail Kravchuk | title=Sur une généralisation des polynomes d'Hermite. | url=http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN | language=French | jfm=55.0799.01 | year=1929 | journal=Comptes Rendus Mathématique | volume=189 | pages=620–622}}
• {{dlmf|id=18.19|title=Hahn Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| last2 = Suslov | first2 = S. K.

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| isbn = 3-540-51123-7

| location = Berlin

| mr = 1149380

| publisher = Springer-Verlag

| series = Springer Series in Computational Physics

| title = Classical Orthogonal Polynomials of a Discrete Variable

| year = 1991}}.

• {{citation

| last = Levenshtein | first = Vladimir I. | author-link = Vladimir Levenshtein

| doi = 10.1109/18.412678

| issue = 5

| journal = IEEE Transactions on Information Theory

| mr = 1366326

| pages = 1303–1321

| title = Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

| volume = 41

| year = 1995}}.

• {{Citation | first1=F. J. | last1=MacWilliams | first2=N. J. A. | last2=Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3 | url-access=registration | url=https://archive.org/details/theoryoferrorcor0000macw }}

{{commons category}}

• [https://web.archive.org/web/20070205055023/http://orthpol.narod.ru/ Krawtchouk Polynomials Home Page]
• [http://mathworld.wolfram.com/KrawtchoukPolynomial.html "Krawtchouk polynomial"] at MathWorld

Category:Orthogonal polynomials