戈特利布多项式

Gottlieb Polynomials

戈特利布多项式是一个以超几何函数定义的正交多项式

${\ell }_n(x,\lambda )=e^{-n\lambda }\sum _k(1-e^{\lambda }{)}^k(\genfrac{}{}{0ex}{}{n}{k})(\genfrac{}{}{0ex}{}{x}{k})=e^{-n\lambda }{}_2{F}_1(-n,-x;1;1-e^{\lambda })$

前面几条戈特利布多项式为：

${\ell }_0(x,\lambda )=1$

${\ell }_1(x,\lambda )=-exp(-\lambda )*(-1-x+x*exp(\lambda ))$

${\ell }_2(x,\lambda )=-(1/2)*exp(-2*\lambda )*(-2-3*x+2*x*exp(\lambda )-x^2+2*x^2*exp(\lambda )-exp(2*\lambda )*x^2+exp(2*\lambda )*x)$

${\ell }_3(x,\lambda )=-(1/6)*exp(-3*\lambda )*(-6-11*x+6*x*exp(\lambda )-6*x^2+9*x^2*exp(\lambda )+3*exp(2*\lambda )*x-x^3+3*x^3*exp(\lambda )-3*exp(2*\lambda )*x^3+exp(3*\lambda )*x^3-3*exp(3*\lambda )*x^2+2*exp(3*\lambda )*x)$

${\ell }_4(x,\lambda )=-(1/24)*exp(-4*\lambda )*(-24-50*x+24*x*exp(\lambda )-35*x^2-exp(4*\lambda )*x^4+4*x^4*exp(\lambda )-6*exp(2*\lambda )*x^4+4*exp(3*\lambda )*x^4+6*exp(4*\lambda )*x-11*exp(4*\lambda )*x^2+6*exp(4*\lambda )*x^3+8*exp(3*\lambda )*x-4*exp(3*\lambda )*x^2+24*x^3*exp(\lambda )-12*exp(2*\lambda )*x^3-8*exp(3*\lambda )*x^3+44*x^2*exp(\lambda )+6*exp(2*\lambda )*x^2+12*exp(2*\lambda )*x-10*x^3-x^4)$

参考文献

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