阿贝尔多项式

数学上的阿贝尔多项式(Abel polynomial)是一个多项式序列，其第n项的形式为

${\displaystyle p_{n}(x)=x(x-an)^{n-1}.}$

该序列以挪威数学家阿贝尔（1802-1829）的名字来命名。

这个多项式为二项型。

例子

当${\displaystyle a=1}$时，此多项式为（OEIS数列A137452）

${\displaystyle p_{0}(x)=1;}$

${\displaystyle p_{1}(x)=x;}$

${\displaystyle p_{2}(x)=-2x+x^{2};}$

${\displaystyle p_{3}(x)=9x-6x^{2}+x^{3};}$

${\displaystyle p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};}$

当${\displaystyle a=1}$ ，此多项式为

${\displaystyle p_{0}(x)=1;}$

${\displaystyle p_{1}(x)=x;}$

${\displaystyle p_{2}(x)=-4x+x^{2};}$

${\displaystyle p_{3}(x)=36x-12x^{2}+x^{3};}$

${\displaystyle p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};}$

${\displaystyle p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};}$

${\displaystyle p_{6}(x)=-248832x+103680x^{2}-17280x^{3}+1440x^{4}-60x^{5}+x^{6};}$

参考文献

• Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. All Polynomials of Binomial Type Are Represented by Abel Polynomials. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 4. 1997, 25 (3–4): 731–738 [2021-01-30]. MR 1655539. Zbl 1003.05011. （原始内容存档于2021-02-08）.

外部链接

• 埃里克·韦斯坦因. Abel Polynomial. MathWorld.