Abel polynomials

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

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

They occur, for instance, in the enumeration of trees (specifically, rooted labeled forests) as well as in connection with geometric probability (the random placement of nonoverlapping arcs on a circle). 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)

$p_0(x)=1;$

$p_1(x)=x;$

$p_2(x)=-2x+x^2;$

$p_3(x)=9x-6x^2+x^3;$

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

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. Series 4 25 (3–4): 731–738. http://www.numdam.org/item?id=ASNSP_1997_4_25_3-4_731_0. 
• Roman, S. (2013). The Umbral Calculus. Dover Publications. pp. 208. ISBN 9780486153421. 

External links

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