William Fleetwood Sheppard

“ In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e. the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry) , i.e. where the value of each term depends on the values of a pair of other quantities. ”

“ The solution, if p1, p2, ... pm are all different, is vn = C1p1n + C2p2n + ... + Cmpmn + Vn, where C1, C2 ... are constants, and vn = Vn is any one solution of the equation. The method of finding a value for Vn depends on the form of N. Certain modifications are required when two or more of the p’s are equal.
It should be observed, in all cases of this kind, that, in describing C1, C2 as “constants,” it is meant that the value of any one, as C1, is the same for all values of n occurring in the series. A “constant” may, however, be a periodic function of n. ”

“ Difference-equations.—The summation of the series ... + un+2 + un-1 + un is a solution of the difference-equation Δvn = un+1, which may also be written (E − 1) vn = un+1. This is a simple form of difference-equation. There are several forms which have been investigated; a simple form, more general than the above, is the linear equation with constant coefficients—
vn+m + a1vn+m-1 + a2vn+m-2 + ... + amvn = N,
where a1, a2, ... am are constants, and N is a given function of n. ”