The Abel polynomials are a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of polynomials named after

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

, defined by the following equation: :

p_n(x)=x(x-an)^

This polynomial sequence is of

binomial type In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities : ...

: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence using

umbral calculus The term umbral calculus has two related but distinct meanings. In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove ...

Examples

For , the polynomials are :

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 , 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

External links

* Polynomials {{polynomial-stub