Abel polynomials

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

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

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. MR 1655539. Zbl 1003.05011.

External links

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