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| The '''Abel polynomials''' in [[mathematics]] form a [[polynomial sequence]], the ''n''th term of which is of the form
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| :<math>p_n(x)=x(x-an)^{n-1}. \,</math>
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| The sequence is named after [[Niels Henrik Abel]] (1802-1829), the Norwegian mathematician.
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| 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]].
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| ==Examples==
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| For {{math|a{{=}}1}}, the polynomials are {{OEIS|A137452}}
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| :<math>p_0(x)=1;</math> | |
| :<math>p_1(x)=x;</math> | |
| :<math>p_2(x)=-2x+x^2;</math>
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| :<math>p_3(x)=9x-6x^2+x^3;</math>
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| :<math>p_4(x)=-64x +48x^2-12x^3+x^4;</math>
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| For {{math|a{{=}}2}}, the polynomials are
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| :<math>p_0(x)=1;</math>
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| :<math>p_1(x)=x;</math>
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| :<math>p_2(x)=-4x+x^2;</math>
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| :<math>p_3(x)=36x-12x^2+x^3;</math>
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| :<math>p_4(x)=-512x +192x^2-24x^3+x^4;</math>
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| :<math>p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5;</math>
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| :<math>p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6;</math>
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| ==References==
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| * {{cite journal | author=Gian-Carlo Rota | authorlink=Gian-Carlo Rota | coauthors=Jianhong (Jackie) Shen, Brian D. Taylor | title=All polynomials of binomial type are represented by Abel polynomials | journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 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}}
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| ==External links==
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| * {{MathWorld | urlname=AbelPolynomial | title=Abel Polynomial}}
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| [[Category:Polynomials]]
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| {{algebra-stub}}
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