Английская Википедия:Abel polynomials

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The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation:

<math>p_n(x)=x(x-an)^{n-1}</math>

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 Шаблон:Math, the polynomials are Шаблон:OEIS

<math>p_0(x)=1;</math>

<math>p_1(x)=x;</math>

<math>p_2(x)=-2x+x^2;</math>

<math>p_3(x)=9x-6x^2+x^3;</math>

<math>p_4(x)=-64x +48x^2-12x^3+x^4;</math>

For Шаблон:Math, the polynomials are

<math>p_0(x)=1;</math>

<math>p_1(x)=x;</math>

<math>p_2(x)=-4x+x^2;</math>

<math>p_3(x)=36x-12x^2+x^3;</math>

<math>p_4(x)=-512x +192x^2-24x^3+x^4;</math>

<math>p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5;</math>

<math>p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6;</math>

References

• Шаблон:Cite journal

External links

• Шаблон:MathWorld

Шаблон:Polynomial-stub