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 Template:Math, 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 Template:Math, the polynomials are

$p_0(x)=1;$

$p_1(x)=x;$

$p_2(x)=-4x+x^2;$

$p_3(x)=36x-12x^2+x^3;$

$p_4(x)=-512x+192x^2-24x^3+x^4;$

$p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5;$

$p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6;$

External links

Template:Polynomial-stub