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 {{math|1=a = 1}}, the polynomials are {{OEIS|A137452}}

: $p_0(x)=1;$

: $p_1(x)=x;$

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

: $p_3(x)=9x-6x^2+x^3;$

: $p_4(x)=-64x +48x^2-12x^3+x^4;$

For {{math|1=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

{{cite journal |last1=Rota |first1=Gian-Carlo | authorlink1=Gian-Carlo Rota |last2=Shen |first2=Jianhong |last3=Taylor |first3=Brian D. |title=All Polynomials of Binomial Type Are Represented by Abel Polynomials |journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |series=Series 4 |volume=25 |issue=3–4 |year=1997 |pages=731–738 |url=http://www.numdam.org/item?id=ASNSP_1997_4_25_3-4_731_0 |mr=1655539 |zbl=1003.05011}}

External links

{{MathWorld | urlname=AbelPolynomial | title=Abel Polynomial}}

Category:Polynomials