These first differences may themselves be differenced, giving the second differences Δu2 - Δu1, Δu3 - Δu2, ..., which are written Δ2u1, Δ2u2,...
Similarly, we form the third differences Δ3u1 = Δ2u2 - Δ2u1, and so on.
As an example, let the original series be the cubes of the natural numbers.
| 1 | 8 | 27 | 64 | 125 | 216 | / 343 | 512 |
| 7 | 19 | 37 | 61 | 91 | / 127 | 169 | |
| 12 | 18 | 24 | 30 | / 36 | 42 | ||
| 6 | 6 | 6 | / 6 | 6 | |||
| 0 | 0 | / 0 | 0 |
Here we begin by writing down the series of cubes as far, say, as 216; beneath these we write the first differences 8 - 1 = 7, 27 - 8 = 19, &c. We thus obtain the part of the table to the left of the diagonal line.
We observe that the third differences are constant, each being 6. (It is easy to prove generally that the nth differences of the series, 1n, 2n, 3n,..., are constant.) Knowing the third differences, we can now extend the table as far as we wish to the right of the diagonal line. We get first 6 + 30 = 36, 36 + 91 = 127, 127 + 216 = 343. We infer that 73 = 343.
Since u1 - u0 = Δu0, we have u1 = u0 + Δu0 = (1 + Δ)u0.
Similarly, u2 = (1 + Δ)u1 = (1 + Δ)2u0; and, generally,
ux = (1 + Δ)xu0
| = | { | 1 + xΔ + | x(x-1) | Δ2 + ... | } | u0 | ||||
| 1×2 |