查看“戈特利布多项式”的源代码


[[File:Gottlieb polynomials.gif|thumb|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]]

:<math>\displaystyle \ell_n(x,\lambda) = e^{-n\lambda}\sum_k(1-e^\lambda)^k\binom{n}{k}\binom{x}{k} =e^{-n\lambda}{}_2F_1(-n,-x;1;1-e^\lambda)</math>


前面几条戈特利布多项式为：

::<math>\displaystyle \ell_0(x,\lambda) =        1             </math>
::<math>\displaystyle \ell_1(x,\lambda) =  -exp(-\lambda)*(-1-x+x*exp(\lambda))                   </math>
::<math>\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)                    </math>
::<math>\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)                    </math>
::<math>\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)                    </math>

==参考文献==


*{{Citation | last1=Gottlieb | first1=M. J. | title=Concerning some polynomials orthogonal on a finite or enumerable set of points. | doi=10.2307/2371307 | jfm=64.0329.01 | year=1938 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=60 | pages=453–458}}
*{{Citation | last1=Rainville | first1=Earl D. | authorlink=Earl Rainville | title=Special functions | publisher=The Macmillan Co. | location=New York | mr=0107725 | year=1960}}

[[Category:正交多项式]]

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