克拉夫楚克多项式

Kravchuk polynomials animation

Kravchuk polynomials animation

克拉夫楚克多项式以超几何函数定义如下

$K_n(x;p,N)={(\genfrac{}{}{0ex}{}{N}{n})}^{-1}\ast$$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{N-x}{n-k})\ast (\genfrac{}{}{0ex}{}{x}{k})\ast (1-1/p{)}^k$=${}_2{F}_1(-n,-x;N;1/p)$

克拉夫楚克多项式的头几项是

$K_0(x,p,N)=1$

$K_1(x,p,N)=1-\frac{x}{p\ast N}$

$K_2(x,p,N)=1+(\frac{-2}{p\ast N}+\frac{1}{(p^2\ast N\ast (-N+1)})\ast x-\frac{x^2}{(p^2\ast N\ast (-N+1))}$

$K_3(x,p,N)=1+(\frac{-3}{(p\ast N)}+\frac{3}{(p^2\ast N\ast (-N+1))}-\frac{2}{(p^3\ast N\ast (-N+1)\ast (-N+2))})\ast x+(-3/(p^2\ast N\ast (-N+1))+3/(p^3\ast N\ast (-N+1)\ast (-N+2)))\ast x^2-x^3/(p^3\ast N\ast (-N+1)\ast (-N+2))$

极限关系

量子Q克拉夫楚克多项式→ 克拉夫楚克多项式：

$\underset{a\to 1}{\lim }=K_n^{qtm}(q^{-}x;p,N;q)=K_n(x;p^{-1},N)$

令双Q克拉夫楚克多项式$c=1=p^{-1}$,并令q→1,即得克拉夫楚克多项式

参考文献

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