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 that this series is ... &Sigma;un-1, &Sigma;un, &Sigma;un+1.... The suffixes are chosen so that we may have &Delta;&Sigma;un = un, whatever n may be; and therefore (§ 4) &Sigma;un may be regarded as being the sum of the terms of the series up to and including un-1. Thus if we write &Sigma;un-1 = C + un-2, where C is any constant, we shall have and so on. This is true whatever C may be, so that the knowledge of ... un-1, un, ... gives us no knowledge of the exact value of &Sigma;un; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of &Sigma;u corresponding to different suffixes.

There is another symbol E, used in conjunction with u to denote the next term in the series. Thus Eun means un+1, so that Eun = un + &Delta;un.

10. Corresponding to the advancing-difference notation there is a receding-difference notation, in which un+1 &minus; un is regarded as a difference of un+1, and may be denoted by &Delta;′un+1, and similarly un+1 &minus; 2un + un-1 may be denoted by &Delta;′²un+1. This notation is only required for certain special purposes, and the usage is not settled (§ 19 (ii.)).

11. The central-difference notation depends on treating un+1 &minus; 2un &minus; un-1 as the second difference of un, and therefore as corresponding to the value xn; but there is no settled system of notation. The following seems to be the most convenient. Since un is a function of xn, and the second difference un+2 &minus; 2un+1 + un is a function of xn+1, the first difference un+1 &minus; un must be regarded as a function of xn+1/2, i.e. of ½(xn + xn+1). We therefore write un+1 &minus; un = &delta;un+1/2, and each difference in the table in § 9 will have the same suffix as the value of x in the same horizontal line; or, if the difference is of an odd order, its suffix will be the means of those of the two nearest values of x. This is shown in the table below.

In this notation, instead of using the symbol E, we use a symbol &mu; to denote the mean of two consecutive values of u, or of two consecutive differences of the same order, the suffixes being assigned on the same principle as in the case of the differences. Thus &mu;un+1/2 = ½(un + un+1, &mu;&delta;un = ½(&delta;un-1/2 + &delta;un+1/2, &c.

If we take the means of the differences of odd order immediately above and below the horizontal line through any value of x, these means, with the differences of even order in that line, constitute the central differences of the corresponding value of u. Thus the table of central differences is as follows, the values obtained as means being placed in brackets to distinguish them from the actual differences:—

Similarly, by taking the means of consecutive values of u and also of consecutive differences of even order, we should get a series of terms and differences central to the intervals xn-2 to xn-1, xn-1 to xn, ....

The terms of the series of which the values of u are the first differences are denoted by &sigma;u, with suffixes on the same principle; the suffixes being chosen so that &delta;&sigma;un shall be equal to un. Thus, if &sigma;un-3/2 = C + un-2, then &sigma;un-1/2 = C + un-2 + un-1, &sigma;n+1/2 = C + un-2 + un-1 + un, &c., and also &mu;&sigma;un-1 = C + un-2 + ½un-1, &mu;&sigma;un = 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 &Delta;, &Sigma;, &delta;, &sigma;, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.

(i.) Instead of treating &Delta;u as a function of x, so that &Delta;un means (&Delta;u)n, we may regard &Delta; as denoting an operation performed on u, and take &Delta;un as meaning &Delta;.un. This applies to the other symbols E, &delta;, &c., whether taken simply or in combination. Thus &Delta;Eun means that we first replace un by un+1, and then replace this by un+2 &minus; un+1.

(ii.) The operations &Delta;, E, &delta;, and &mu;, whether performed separately or in combination, or in combination also with numerical multipliers and with the operation of differentiation denoted by D (&equiv; d/dx), follow the ordinary rules of algebra: e.g. &Delta;(un + vn) = &Delta;un + &Delta;vn, &Delta;Dun = D&Delta;un, &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 + &Delta;un = 1·un + &Delta;·un, so that we may write E = 1 + &Delta;, or &Delta; = E &minus; 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 + &Delta;)n, &Delta;n = (E-1)n, and expand by the binomial theorem and then operate on u0, we get the general formulae $u_{n} = u_{0} + n\Delta u_{0} + \frac{n\cdot n - 1}{1\cdot 2}\Delta^{2}u_{0} + \ldots + \Delta^{n}u_{0},$ $\Delta^{n}u_{0} = u_{n} - nu_{n-1} + \frac{n\cdot n - 1}{1\cdot 2}u_{n-2} + \ldots + (-1)^{n}u_{0},$ 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 &Sigma; and &sigma;, and to the operation which represents integration. The latter is sometimes denoted by D-1; and, since &Delta;&Sigma;un = un, and &delta;&sigma;un = un, we might similarly replace &Sigma; and &sigma; by &Delta;-1 and &delta;-1. These symbols can be combined with &Delta;, 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, &Delta;-1, &delta;-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 &Delta;vr = ur for all integral values of r, then the sum of the terms um, um+1, ... un is vn+1 &minus; 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 &Delta;ur, is a rational integral function of r of degree p &minus; 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 &Delta; · (r + a + 1) (r + a + 2) ... (r + b) = (b &minus; a)·(r + a + 2) ... (r + b), whence, changing a into a-1, &Sigma;(r + a + 1) (r + a + 2) ... (r + b) = const. + (r + a)(r + a + 1) ... (r + b)/(b &minus; 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).

(iii.) Any rational integral function can be converted into the sum of a number of factorials; and thus the sum of a series of which such a function is the general term can be found. For example, it may be shown in this way that the sum of the pth powers of the first n natural numbers is a rational integral function of n of degree p + 1, the coefficient of np+1 being 1/(p + 1).

15. Difference-equations.—The summation of the series ... + un+2 + un-1 + un is a solution of the difference-equation &Delta;vn = un+1, which may also be written (E &minus; 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. This may be written (Em + a1Em-1 + ... + am)vn = N or (E &minus; p1)(E &minus; p2) ... (E &minus; pm)vn = N. 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.

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.)