μσun-1 = C + un-2 + ½un-1, μσun = C + un-2 + un-1 + ½un, &c.,
C being an arbitrary constant which must remain the same throughout any series of operations.
Operators and Symbolic Methods.
12. There are two further stages in the use of the symbols Δ, Σ, δ, σ, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.
(i.) Instead of treating Δu as a function of x, so that Δun means (Δu)n, we may regard Δ as denoting an operation performed on u, and take Δun as meaning Δ.un. This applies to the other symbols E, δ, &c., whether taken simply or in combination. Thus ΔEun means that we first replace un by un+1, and then replace this by un+2 − un+1.
(ii.) The operations Δ, E, δ, and μ, whether performed separately or in combination, or in combination also with numerical multipliers and with the operation of differentiation denoted by D (≡ d/dx), follow the ordinary rules of algebra: e.g. Δ(un + vn) = Δun + Δvn, ΔDun = DΔun, &c. Hence the symbols can be separated from the functions on which the operations are performed, and treated as if they were algebraical quantities. For instance, we have
E·un = un+1 = un + Δun = 1·un + Δ·un,
so that we may write E = 1 + Δ, or Δ = E − 1. The first of these is nothing more than a statement, in concise form, that if we take two quantities, subtract the first from the second, and add the result to the first, we get the second. This seems almost a truism. But, if we deduce En = (1 + Δ)n, Δn = (E-1)n, and expand by the binomial theorem and then operate on u0, we get the general formulae
| un = u0 + nΔu0 + | n·n − 1 | Δ2u0 + ... + Δnu0, |
| 1·2 |
| Δnu0 = un − nun-1 + | n·n − 1 | un-2 + ... + (-1)nu0, |
| 1·2 |