TOPICS

# Actuarial Polynomial

The polynomials given by the Sheffer sequence with

 (1) (2)

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

 (5) (6)

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

 (9) (10) (11) (12)

Sheffer Sequence

## Explore with Wolfram|Alpha

More things to try:

## 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