Actuarial polynomials

In mathematics, the actuarial polynomials a^(β)
_n(x) are polynomials studied by Toscano (1950) given by the generating function

\displaystyle \sum _{n}{\frac {a_{n}^{(\beta )}(x)}{n!}}t^{n}=\exp(\beta t+x(1-e^{t}))

References

Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., vol. 19, Berlin, New York: Springer-Verlag, ISBN 9783540031239, MR 0094466
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185 Reprinted by Dover, 2005
Toscano, Letterio (1950), "Una classe di polinomi della matematica attuariale", Rivista di Matematica della Università di Parma (in Italian), 1: 459–470, MR 0040480, Zbl 0040.03204