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
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)
Explore with Wolfram|Alpha
WolframAlpha
More things to try:
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 PolynomialCite this as:
Weisstein, Eric W. "Abel Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelPolynomial.html