Expansion of Functions and Forward Differences

In Sections 13-15, Servois describes what he calls “the general theory of the expansion of functions into series,” which includes both the Newton forward difference formula and Taylor series. In Section 13, Servois introduces his equation (25), a very general formula for the expansion of a function \(F\) in terms of a sequence of auxiliary functions \(\varphi, \varphi^{\prime}, \varphi^{\prime\prime}, \ldots\). In order to establish series (25) “analytically,” Servois uses a generalized notion of the difference quotient, as given in his formula (24). Then in Section 14, he uses similar techniques to derive the Newton forward difference formula, his (36).

In Section 15, Servois makes the transition from forward differences to powers of a variable. He lets \[\varphi (x) = x - p, \; \varphi^{\prime} (x) = x - p - \alpha, \; \varphi^{\prime\prime} (x) = x - p - 2\alpha, \; \ldots\] for a fixed number \(p\) and a fixed increment \(\alpha\). At the top of his page [108], he observes, but does not clearly explain, that \[\Pi_n = \frac{(x - p)(x - p - \alpha) \cdots (x - p - (n - 1)\alpha)}{\alpha^n}\] can be written in the form \[\Sigma_n = a_1 \left(\frac{x - p}{\alpha}\right) + a_2 \left(\frac{x - p}{\alpha}\right)^2 + \ldots + a_n \left(\frac{x - p}{\alpha}\right)^n.\] Here, we have used subscript notation where Servois used \(A\), \(B\), \(C\), \(\ldots\). We have also introduced the notation \(\Pi_n\) and \(\Sigma_n\). It is easy to prove Servois’ claim by induction, because \[\Pi_{n+1} = \Pi_{n} \left(\frac{x-p - n\alpha}{\alpha}\right)\] \[\phantom{xxxxx}= \Sigma_n \left[\left(\frac{x-p}{\alpha}\right) - n \right],\] which has the form \(\Sigma_{n+1}\).

Servois uses this observation to rewrite his series (33), which is a form of the Newton forward difference formula, in terms of the powers of \[\frac{x - p}{\alpha}.\]