TOPICS
Search

Actuarial Polynomial

The polynomials a_n^((beta))(x) given by the Sheffer sequence with

g(t)=(1-t)^(-beta)
(1)
f(t)=ln(1-t),
(2)

giving generating function

sum_(k=0)^infty(a_n^((beta)))/(k!)t^k=e^(x(1-e^t)+betat).
(3)

The Sheffer identity is

a_n^((beta))(x+y)=sum_(k=0)^n(n; k)a_k^((beta))(y)phi_(n-k)(-x),
(4)

where phi_n(x) is a Bell polynomial. The actuarial polynomials are given in terms of the Bell polynomials phi_n(x) by

a_n^((beta))(x)=(1-t)^betaphi_n(-x)
(5)
=sum_(k=0)^(n)(beta; k)phi_n^((k))(-x).
(6)

They are related to the Stirling numbers of the second kind S(n,m) by

a_n^((beta))(x)=sum_(k=0)^n(beta; k)sum_(j=k)^nS(n,j)(j)_k(-x)^(j-k),
(7)

where (n; k) is a binomial coefficient and (x)_n is a falling factorial. The actuarial polynomials also satisfy the identity

a_n^((beta))(-x)=e^(-x)sum_(k=0)^infty((k+beta)^n)/(k!)x^k
(8)

(Roman 1984, p. 125; Whittaker and Watson 1990, p. 336).

The first few polynomials are

a_0^((beta))(x)=1
(9)
a_1^((beta))(x)=-x+beta
(10)
a_2^((beta))(x)=x^2-x(1+2beta)+beta^2
(11)
a_3^((beta))(x)=-x^3+3x^2(beta+1)-x(3beta^2+3beta+1)+beta^3.
(12)

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