Abel polynomials

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

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

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

 
 
 
 
 

For a = 2, the polynomials are

 
 
 
 
 
 
 

References

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  • Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. (1997). "All Polynomials of Binomial Type Are Represented by Abel Polynomials". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. Series 4. 25 (3–4): 731–738. MR 1655539. Zbl 1003.05011.
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