Abel polynomials explained

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

	n-1
p
	n(x)=x(x-an)

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, the polynomials are

p_0(x)=1;

p_1(x)=x;

	2;
p
	2(x)=-2x+x

	2+x
p
	3(x)=9x-6x

^3;

p_4(x)=-64x+48x^2-12x^3+x^4;

For, the polynomials are

p_0(x)=1;

p_1(x)=x;

	2;
p
	2(x)=-4x+x

	2+x
p
	3(x)=36x-12x

^3;

p_4(x)=-512x+192x^2-24x^3+x^4;

	2+600x
p
	5(x)=10000x-4000x

^3-40x^4+x^5;

	2-17280x
p
	6(x)=-248832x+103680x

^3+1440x^4-60x^5+x^6;

References

Rota . Gian-Carlo . Gian-Carlo Rota . 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 . Series 4 . 25 . 3–4 . 1997 . 731–738 . 1655539 . 1003.05011.