戈特利布多项式维基百科,自由的 encyclopedia 戈特利布多项式是一个以超几何函数定义的正交多项式 ℓ n ( x , λ ) = e − n λ ∑ k ( 1 − e λ ) k ( n k ) ( x k ) = e − n λ 2 F 1 ( − n , − x ; 1 ; 1 − e λ ) {\displaystyle \displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}(1-e^{\lambda })^{k}{\binom {n}{k}}{\binom {x}{k}}=e^{-n\lambda }{}_{2}F_{1}(-n,-x;1;1-e^{\lambda })} Gottlieb Polynomials 前面几条戈特利布多项式为: ℓ 0 ( x , λ ) = 1 {\displaystyle \displaystyle \ell _{0}(x,\lambda )=1} ℓ 1 ( x , λ ) = − e x p ( − λ ) ∗ ( − 1 − x + x ∗ e x p ( λ ) ) {\displaystyle \displaystyle \ell _{1}(x,\lambda )=-exp(-\lambda )*(-1-x+x*exp(\lambda ))} ℓ 2 ( x , λ ) = − ( 1 / 2 ) ∗ e x p ( − 2 ∗ λ ) ∗ ( − 2 − 3 ∗ x + 2 ∗ x ∗ e x p ( λ ) − x 2 + 2 ∗ x 2 ∗ e x p ( λ ) − e x p ( 2 ∗ λ ) ∗ x 2 + e x p ( 2 ∗ λ ) ∗ x ) {\displaystyle \displaystyle \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)} ℓ 3 ( x , λ ) = − ( 1 / 6 ) ∗ e x p ( − 3 ∗ λ ) ∗ ( − 6 − 11 ∗ x + 6 ∗ x ∗ e x p ( λ ) − 6 ∗ x 2 + 9 ∗ x 2 ∗ e x p ( λ ) + 3 ∗ e x p ( 2 ∗ λ ) ∗ x − x 3 + 3 ∗ x 3 ∗ e x p ( λ ) − 3 ∗ e x p ( 2 ∗ λ ) ∗ x 3 + e x p ( 3 ∗ λ ) ∗ x 3 − 3 ∗ e x p ( 3 ∗ λ ) ∗ x 2 + 2 ∗ e x p ( 3 ∗ λ ) ∗ x ) {\displaystyle \displaystyle \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)} ℓ 4 ( x , λ ) = − ( 1 / 24 ) ∗ e x p ( − 4 ∗ λ ) ∗ ( − 24 − 50 ∗ x + 24 ∗ x ∗ e x p ( λ ) − 35 ∗ x 2 − e x p ( 4 ∗ λ ) ∗ x 4 + 4 ∗ x 4 ∗ e x p ( λ ) − 6 ∗ e x p ( 2 ∗ λ ) ∗ x 4 + 4 ∗ e x p ( 3 ∗ λ ) ∗ x 4 + 6 ∗ e x p ( 4 ∗ λ ) ∗ x − 11 ∗ e x p ( 4 ∗ λ ) ∗ x 2 + 6 ∗ e x p ( 4 ∗ λ ) ∗ x 3 + 8 ∗ e x p ( 3 ∗ λ ) ∗ x − 4 ∗ e x p ( 3 ∗ λ ) ∗ x 2 + 24 ∗ x 3 ∗ e x p ( λ ) − 12 ∗ e x p ( 2 ∗ λ ) ∗ x 3 − 8 ∗ e x p ( 3 ∗ λ ) ∗ x 3 + 44 ∗ x 2 ∗ e x p ( λ ) + 6 ∗ e x p ( 2 ∗ λ ) ∗ x 2 + 12 ∗ e x p ( 2 ∗ λ ) ∗ x − 10 ∗ x 3 − x 4 ) {\displaystyle \displaystyle \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})}
戈特利布多项式是一个以超几何函数定义的正交多项式 ℓ n ( x , λ ) = e − n λ ∑ k ( 1 − e λ ) k ( n k ) ( x k ) = e − n λ 2 F 1 ( − n , − x ; 1 ; 1 − e λ ) {\displaystyle \displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}(1-e^{\lambda })^{k}{\binom {n}{k}}{\binom {x}{k}}=e^{-n\lambda }{}_{2}F_{1}(-n,-x;1;1-e^{\lambda })} Gottlieb Polynomials 前面几条戈特利布多项式为: ℓ 0 ( x , λ ) = 1 {\displaystyle \displaystyle \ell _{0}(x,\lambda )=1} ℓ 1 ( x , λ ) = − e x p ( − λ ) ∗ ( − 1 − x + x ∗ e x p ( λ ) ) {\displaystyle \displaystyle \ell _{1}(x,\lambda )=-exp(-\lambda )*(-1-x+x*exp(\lambda ))} ℓ 2 ( x , λ ) = − ( 1 / 2 ) ∗ e x p ( − 2 ∗ λ ) ∗ ( − 2 − 3 ∗ x + 2 ∗ x ∗ e x p ( λ ) − x 2 + 2 ∗ x 2 ∗ e x p ( λ ) − e x p ( 2 ∗ λ ) ∗ x 2 + e x p ( 2 ∗ λ ) ∗ x ) {\displaystyle \displaystyle \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)} ℓ 3 ( x , λ ) = − ( 1 / 6 ) ∗ e x p ( − 3 ∗ λ ) ∗ ( − 6 − 11 ∗ x + 6 ∗ x ∗ e x p ( λ ) − 6 ∗ x 2 + 9 ∗ x 2 ∗ e x p ( λ ) + 3 ∗ e x p ( 2 ∗ λ ) ∗ x − x 3 + 3 ∗ x 3 ∗ e x p ( λ ) − 3 ∗ e x p ( 2 ∗ λ ) ∗ x 3 + e x p ( 3 ∗ λ ) ∗ x 3 − 3 ∗ e x p ( 3 ∗ λ ) ∗ x 2 + 2 ∗ e x p ( 3 ∗ λ ) ∗ x ) {\displaystyle \displaystyle \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)} ℓ 4 ( x , λ ) = − ( 1 / 24 ) ∗ e x p ( − 4 ∗ λ ) ∗ ( − 24 − 50 ∗ x + 24 ∗ x ∗ e x p ( λ ) − 35 ∗ x 2 − e x p ( 4 ∗ λ ) ∗ x 4 + 4 ∗ x 4 ∗ e x p ( λ ) − 6 ∗ e x p ( 2 ∗ λ ) ∗ x 4 + 4 ∗ e x p ( 3 ∗ λ ) ∗ x 4 + 6 ∗ e x p ( 4 ∗ λ ) ∗ x − 11 ∗ e x p ( 4 ∗ λ ) ∗ x 2 + 6 ∗ e x p ( 4 ∗ λ ) ∗ x 3 + 8 ∗ e x p ( 3 ∗ λ ) ∗ x − 4 ∗ e x p ( 3 ∗ λ ) ∗ x 2 + 24 ∗ x 3 ∗ e x p ( λ ) − 12 ∗ e x p ( 2 ∗ λ ) ∗ x 3 − 8 ∗ e x p ( 3 ∗ λ ) ∗ x 3 + 44 ∗ x 2 ∗ e x p ( λ ) + 6 ∗ e x p ( 2 ∗ λ ) ∗ x 2 + 12 ∗ e x p ( 2 ∗ λ ) ∗ x − 10 ∗ x 3 − x 4 ) {\displaystyle \displaystyle \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})}