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Abel Polynomial

A polynomial A_n(x;a) given by the associated Sheffer sequence with

f(t)=te^(at),
(1)

given by

A_n(x;a)=x(x-an)^(n-1).
(2)

The generating function is

sum_(k=0)^infty(A_k(x;a))/(k!)t^k=e^(xW(at)/a),
(3)

where W(x) is the Lambert W-function. The associated binomial identity is

(x+y)(x+y-an)^(n-1)=sum_(k=0)^n(n; k)xy(x-ak)^(k-1)[y-a(n-k)]^(n-k-1),
(4)

where (n; k) is a binomial coefficient, a formula originally due to Abel (Riordan 1979, p. 18; Roman 1984, pp. 30 and 73).

The first few Abel polynomials are

A_0(x;a)=1
(5)
A_1(x;a)=x
(6)
A_2(x;a)=x(x-2a)
(7)
A_3(x;a)=x(x-3a)^2
(8)
A_4(x;a)=x(x-4a)^3.
(9)

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References

Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979.Roman, S. "The Abel Polynomials." §4.1.5 in The Umbral Calculus. New York: Academic Press, pp. 29-30 and 72-75, 1984.

Referenced on Wolfram|Alpha

Abel Polynomial

Cite this as:

Weisstein, Eric W. "Abel Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelPolynomial.html

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