查看“︁克拉夫楚克多项式”︁的源代码

{{noteTA
|1=zh-cn:克拉夫楚克; zh-tw:克拉夫丘克
}}
[[File:Kravchuk 1.gif|thumb|Kravchuk polynomials  animation]]
[[File:Kravchuk 2.gif|thumb|Kravchuk polynomials  animation]]
'''克拉夫楚克多项式'''以[[超几何函数]]定义如下

<math>K_n(x;p,N)={N \choose n}^{-1}*</math><math>\sum_{k=0}^{n}{N-x \choose n-k}*{x \choose k}*(1-1/p)^k</math>=<math>_2F_1(-n,-x;N;1/p)</math>


克拉夫楚克多项式的头几项是

<math>K_0(x,p,N)= 1</math>

<math>K_1(x,p,N)= 1-\frac{x}{p*N}   </math>

<math> K_2(x,p,N)= 1+(\frac{-2}{p*N}+\frac{1}{(p^2*N*(-N+1)})*x-\frac{x^2}{(p^2*N*(-N+1))}  </math>

<math>K_3(x,p,N)= 1+(\frac{-3}{(p*N)}+\frac{3}{(p^2*N*(-N+1))}-\frac{2}{(p^3*N*(-N+1)*(-N+2))})*x+(-3/(p^2*N*(-N+1))+3/(p^3*N*(-N+1)*(-N+2)))*x^2-x^3/(p^3*N*(-N+1)*(-N+2))   </math>

==极限关系==
[[量子Q克拉夫楚克多项式]]→ '''克拉夫楚克多项式'''：

<math>\lim_{a \to 1}=K_{n}^{qtm}(q^-{x};p,N;q)=K_{n}(x;p^{-1},N)</math>

令[[双Q克拉夫楚克多项式]]<math>c=1=p^{-1}</math>,并令q→1,即得'''克拉夫楚克多项式'''

==参考文献==
<references/>
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*{{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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*{{citation
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*{{Citation | author=F. J. MacWilliams | coauthors=N. J. A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3}}

[[Category:正交多项式]]