n
natural numbers is a rational integral function of n of degree p+1,
the coefficient of n^p+1 being 1/(p+1).

15. _Difference-equations._--The summation of the series ... + u_(n+2)
+ u_(n-1) + u_n is a solution of the _difference-equation_ [Delta]v_n
= u_(n+1), which may also be written (E-1)v_n = u_(n+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_--

v_(n+m) + a1v_(n+m-1) + a2v_(n+m-2) + ... + a_mv_n = N,

where a1, a2, ... am are constants, and N is a given function of n.
This may be written

(E^m + a1E^(m-1) + ... + a_m)v_n = N

or

(E-p1)(E-p2) ... (E-p_m)v_n = N.

The solution, if p1, p2, ... pm are all different, is vn = C1p1^n +
C2p2^n + ... + C_mp_m^n + V_n, where C1, C2 ... are constants, and v_n
= V_n 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.

_Applications to Continuous Functions._

16. The cases of greatest practical importance are those in which u is
a continuous function of x. The terms u1, u2 ... of the series then
represent the successive values of u corresponding to x = x1, x2....
The important applications of the theory in these cases are to (i.)
relations between differences and differential coefficients, (ii.)
interpolation, or the determination of intermediate values of u, and
(iii.) relations between sums and integrals.

17. Starting from any pair of values x0 and u0, we may suppose the
interval h from x0 to x1 to be divided into q equal portions. If we
suppose the corresponding values of u to be obtained, and their
differences taken, the successive advancing differences of u0 being
denoted by dPu0, dP^2u0 ..., we have (S 3 (ii.))

q.q - 1
u1 = u0 + qdPu0 + ------- dP^2u0 + ....
1.2

When q is made indefinitely great, this (writing f(x) for u) becomes
Taylor's Theorem (INFINITESIMAL CALCULUS)