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 p n ( x ) = x ( x − a n ) n − 1 . {\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 a = 1 {\displaystyle a=1} 时，此多项式为（OEIS数列A137452） p 0 ( x ) = 1 ; {\displaystyle p_{0}(x)=1;} p 1 ( x ) = x ; {\displaystyle p_{1}(x)=x;} p 2 ( x ) = − 2 x + x 2 ; {\displaystyle p_{2}(x)=-2x+x^{2};} p 3 ( x ) = 9 x − 6 x 2 + x 3 ; {\displaystyle p_{3}(x)=9x-6x^{2}+x^{3};} p 4 ( x ) = − 64 x + 48 x 2 − 12 x 3 + x 4 ; {\displaystyle p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};} 当 a = 1 {\displaystyle a=1} ，此多项式为 p 0 ( x ) = 1 ; {\displaystyle p_{0}(x)=1;} p 1 ( x ) = x ; {\displaystyle p_{1}(x)=x;} p 2 ( x ) = − 4 x + x 2 ; {\displaystyle p_{2}(x)=-4x+x^{2};} p 3 ( x ) = 36 x − 12 x 2 + x 3 ; {\displaystyle p_{3}(x)=36x-12x^{2}+x^{3};} p 4 ( x ) = − 512 x + 192 x 2 − 24 x 3 + x 4 ; {\displaystyle p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};} p 5 ( x ) = 10000 x − 4000 x 2 + 600 x 3 − 40 x 4 + x 5 ; {\displaystyle p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};} p 6 ( x ) = − 248832 x + 103680 x 2 − 17280 x 3 + 1440 x 4 − 60 x 5 + x 6 ; {\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.
