Actuarial polynomials

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

[math]\displaystyle{ \displaystyle \sum_n \frac{a_n^{(\beta)}(x)}{n!}t^n = \exp(\beta t +x(1-e^t)) }[/math]

(Roman 1984), (Boas Buck).

References

Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., 19, Berlin, New York: Springer-Verlag, ISBN 9783540031239, https://books.google.com/books?id=eihMuwkh4DsC
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, https://books.google.com/books?id=JpHjkhFLfpgC Reprinted by Dover, 2005
Toscano, Letterio (1950), "Una classe di polinomi della matematica attuariale" (in Italian), Rivista di Matematica della Università di Parma 1: 459–470