The polynomials
given by the Sheffer sequence with
giving generating function
|
(3)
|
The Sheffer identity is
|
(4)
|
where
is a Bell polynomial. The actuarial polynomials
are given in terms of the Bell polynomials by
They are related to the Stirling numbers of the second kind by
|
(7)
|
where
is a binomial coefficient and is a falling factorial.
The actuarial polynomials also satisfy the identity
|
(8)
|
(Roman 1984, p. 125; Whittaker and Watson 1990, p. 336).
The first few polynomials are
See also
Sheffer Sequence
Explore with Wolfram|Alpha
References
Boas, R. P. and Buck, R. C. Polynomial Expansions of Analytic Functions, 2nd print., corr. New York: Academic Press,
p. 42, 1964.Erdélyi, A.; Magnus, W.; Oberhettinger, F.;
and Tricomi, F. G. Higher
Transcendental Functions, Vol. 3. New York: Krieger, p. 254, 1981.Roman,
S. "The Actuarial Polynomial." §4.3.4 in The
Umbral Calculus. New York: Academic Press, pp. 123-125, 1984.Whittaker,
E. T. and Watson, G. N. A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, 1990.Referenced on Wolfram|Alpha
Actuarial Polynomial
Cite this as:
Weisstein, Eric W. "Actuarial Polynomial."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ActuarialPolynomial.html
Subject classifications