On a class of polynomials and their application to actuarial problems

Abstract

1. In a paper 1 “On the Mathematical Foundations of Actuarial Science.” Quoted below as “M. F.A.S.”. read before the Sixth Scandinavian Congress of Mathematicians (Copenhagen 1925) I have derived the remainder terms of several formulas, frequently used in actuarial mathematics, and shown how they may be applied if the mortality table follows Makeham' Law. These remainder terms depend mostly on the derivative of a certain order of the function lx , and were obtained by successive derivations of the relation eliminating, after each derivation, l′x by this same relation. It was shown that if the table follows Makeham' Law, is a polynomial of degree n in µx, and these polynomials were calculated as far as n = 5. They rapidly become complicated; and from the point of view I have adopted in the paper quoted above, it becomes a problem of some importance to find simple and yet sufficiently accurate limits to in the general case, that is, for an arbitrary value of n. In order to do this, it will be suitable to begin with a brief in vestigation of a certain class of polynomials of a general type which deserve attention also for their own sake. We do not strictly confine ourselves to the properties which are of immediate practical utility.