Abel polynomials

The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation:

p n ( x ) = x ( x − a n ) n − 1 {\displaystyle p_{n}(x)=x(x-an)^{n-1}}

This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence using umbral calculus.

Examples

For a = 1, the polynomials are (sequence A137452 in the OEIS)

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};}

For a = 2, the polynomials are

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};}

References

  • Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. (1997). "All Polynomials of Binomial Type Are Represented by Abel Polynomials". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. Series 4. 25 (3–4): 731–738. MR 1655539. Zbl 1003.05011.
  • Weisstein, Eric W. "Abel Polynomial". MathWorld.


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