SOME UMBRAL CHARACTERISTICS OF THE ACTUARIAL POLYNOMIALS

Abstract

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and sub- sets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponen- tial generating functions. The actuarial polynomials a ( ) n (x); n = 0; 1; 2; , which was suggested by Toscano, have the following ex- ponential generating function: 1 X n=0 a ( ) n (x) n! t n = exp(t +x(1 e t )): A linear functional on polynomial space can be identied with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplica- tion. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.