Abel polynomials

The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

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

The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.

This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence in the 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

$\displaystyle{ p_0(x)=1; }$
$\displaystyle{ p_1(x)=x; }$
$\displaystyle{ p_2(x)=-4x+x^2; }$
$\displaystyle{ p_3(x)=36x-12x^2+x^3; }$
$\displaystyle{ p_4(x)=-512x +192x^2-24x^3+x^4; }$
$\displaystyle{ p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5; }$
$\displaystyle{ p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6; }$