Abel polynomials

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

${\displaystyle p_{n}(x)=x(x-an)^{n-1}}$

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 )

${\displaystyle p_{0}(x)=1;}$

${\displaystyle p_{1}(x)=x;}$

${\displaystyle p_{2}(x)=-2x+x^{2};}$

${\displaystyle p_{3}(x)=9x-6x^{2}+x^{3};}$

${\displaystyle p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};}$

For a = 2, the polynomials are

${\displaystyle p_{0}(x)=1;}$

${\displaystyle p_{1}(x)=x;}$

${\displaystyle p_{2}(x)=-4x+x^{2};}$

${\displaystyle p_{3}(x)=36x-12x^{2}+x^{3};}$

${\displaystyle p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};}$

${\displaystyle p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};}$

${\displaystyle p_{6}(x)=-248832x+103680x^{2}-17280x^{3}+1440x^{4}-60x^{5}+x^{6};}$

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.
• Roman, S. (2013). The Umbral Calculus. Dover Publications. p. 208. ISBN   9780486153421.

External links

• Weisstein, Eric W. "Abel Polynomial". MathWorld .