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The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

$$p_n(x)=x(x-an)^{n-1}. \,$$

The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.

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

Examples

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

$$p_0(x)=1;$$
$$p_1(x)=x;$$
$$p_2(x)=-2x+x^2;$$
$$p_3(x)=9x-6x^2+x^3;$$
$$p_4(x)=-64x +48x^2-12x^3+x^4;$$

For a=2, the polynomials are

$$p_0(x)=1;$$
$$p_1(x)=x;$$
$$p_2(x)=-4x+x^2;$$
$$p_3(x)=36x-12x^2+x^3;$$
$$p_4(x)=-512x +192x^2-24x^3+x^4;$$
$$p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5;$$
$$p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6;$$

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 Sér. 4 25 (3–4): 731–738. MR 1655539. Zbl 1003.05011.