Abel polynomials

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

[math]\displaystyle{ p_n(x)=x(x-an)^{n-1} }[/math]

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)

[math]\displaystyle{ p_0(x)=1; }[/math]

[math]\displaystyle{ p_1(x)=x; }[/math]

[math]\displaystyle{ p_2(x)=-2x+x^2; }[/math]

[math]\displaystyle{ p_3(x)=9x-6x^2+x^3; }[/math]

[math]\displaystyle{ p_4(x)=-64x +48x^2-12x^3+x^4; }[/math]

For a = 2, the polynomials are

[math]\displaystyle{ p_0(x)=1; }[/math]

[math]\displaystyle{ p_1(x)=x; }[/math]

[math]\displaystyle{ p_2(x)=-4x+x^2; }[/math]

[math]\displaystyle{ p_3(x)=36x-12x^2+x^3; }[/math]

[math]\displaystyle{ p_4(x)=-512x +192x^2-24x^3+x^4; }[/math]

[math]\displaystyle{ p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5; }[/math]

[math]\displaystyle{ p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6; }[/math]

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. http://www.numdam.org/item?id=ASNSP_1997_4_25_3-4_731_0. 

External links

• Weisstein, Eric W.. "Abel Polynomial". http://mathworld.wolfram.com/AbelPolynomial.html.