戈特利布多項式

Gottlieb Polynomials

戈特利布多項式是一個以超幾何函數定義的正交多項式

$\displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}(1-e^{\lambda })^{k}{\binom {n}{k}}{\binom {x}{k}}=e^{-n\lambda }{}_{2}F_{1}(-n,-x;1;1-e^{\lambda })$

前面幾條戈特利布多項式為：

$\displaystyle \ell _{0}(x,\lambda )=1$

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

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

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

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

參考文獻

• Gottlieb, M. J., Concerning some polynomials orthogonal on a finite or enumerable set of points., American Journal of Mathematics, 1938, 60: 453–458, ISSN 0002-9327, JFM 64.0329.01, doi:10.2307/2371307 
• Rainville, Earl D., Special functions, New York: The Macmillan Co., 1960, MR 0107725