The Abel polynomials in mathematics form a

polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in e ...

, the ''n''th term of which is of the form :

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

The sequence is 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 ...

(1802-1829), the Norwegian mathematician. 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 :p ...

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

umbral calculus In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Bliss ...

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 {{algebra-stub