Abel polynomials - Alchetron, The Free Social Encyclopedia

The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

p n ( x ) = x ( x − a n ) n − 1 .

The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.

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.

Examples

For a=1, the polynomials are (sequence A137452 in the OEIS)

p 0 ( x ) = 1 ; p 1 ( x ) = x ; p 2 ( x ) = − 2 x + x 2 ; p 3 ( x ) = 9 x − 6 x 2 + x 3 ; p 4 ( x ) = − 64 x + 48 x 2 − 12 x 3 + x 4 ;

For a=2, the polynomials are

p 0 ( x ) = 1 ; p 1 ( x ) = x ; p 2 ( x ) = − 4 x + x 2 ; p 3 ( x ) = 36 x − 12 x 2 + x 3 ; p 4 ( x ) = − 512 x + 192 x 2 − 24 x 3 + x 4 ; p 5 ( x ) = 10000 x − 4000 x 2 + 600 x 3 − 40 x 4 + x 5 ; p 6 ( x ) = − 248832 x + 103680 x 2 − 17280 x 3 + 1440 x 4 − 60 x 5 + x 6 ;

References

Abel polynomials Wikipedia

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