which are identical with the formulae in (ii.) and (i.) of § 3.
(iii.) What has been said under (ii.) applies, with certain reservations, to the operations Σ and σ, and to the operation which represents integration. The latter is sometimes denoted by D-1; and, since ΔΣun = un, and δσun = un, we might similarly replace Σ and σ by Δ-1 and δ-1. These symbols can be combined with Δ, E, &c. according to the ordinary laws of algebra, provided that proper account is taken of the arbitrary constants introduced by the operations D-1, Δ-1, δ-1.
Applications to Algebraical Series.
13. Summation of Series.—If ur, denotes the (r + 1)th term of a series, and if vr is a function of r such that Δvr = ur for all integral values of r, then the sum of the terms um, um+1, ... un is vn+1 − vm. Thus the sum of a number of terms of a series may often be found by inspection, in the same kind of way that an integral is found.
14. Rational Integral Functions.—(i.) If ur is a rational integral function of r of degree p, then Δur, is a rational integral function of r of degree p − 1.
(ii.) A particular case is that of a factorial, i.e. a product of the form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding factor by 1. We have
Δ · (r + a + 1) (r + a + 2) ... (r + b) = (b − a)·(r + a + 2) ... (r + b),
whence, changing a into a-1,
Σ(r + a + 1) (r + a + 2) ... (r + b) = const. + (r + a)(r + a + 1) ... (r + b)/(b − a + 1).
A similar method can be applied to the series whose (r + 1)th term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).