5. As an example, take the arithmetical series_Encyclopaedia Britannica, 11th Edition, "Diameter" to "Dinarchus" Volume 8, Slice 4

a, a + p, a + 2p, ...

The first differences are p, p, p, ... and the differences of any

higher order are zero. Hence, by (ii.) of S 3, the (n + 1)th term is a

+ np, and, by S 4, the sum of the first n terms is na + 1/2n(n - 1)p =

1/2n{2a + (n - 1)p}.

6 As another example, take the series 1, 8, 27, ... the terms of which

are the cubes of 1, 2, 3, ... The first, second and third differences

of the first term are 7, 12 and 6, and it may be shown (S 14 (i.))

that all differences of a higher order are zero. Hence the sum of the

first n terms is

n.n - 1 n.n - 1.n - 2 n.n - 1.n - 2.n - 3

n + 7 ------- + 12 ------------- + 6 ------------------- =

1.2 1.2.3 1.2.3.4

1/4n^4 + 1/2n^3 + 1/4n^2 = {1/2n(n + 1)}^2.

7. In S 3 we have described b - a, c - 2b + a, ... as the first,

second, ... differences of a. This ascription of the differences to

particular terms of the series is quite arbitrary. If we read the

differences in the table of S 2 upwards to the right instead of

downwards to the right, we might describe e - d, e - 2d + c, ... as

the first, second, ... differences of e. On the other hand, the term

of greatest weight in c -2b + a, i.e. the term which has the

numerically greatest coefficient, is b, and therefore c - 2b + a might

properly be regarded as the second difference of b, and similarly e -

4d + 6c - 4b + a might be regarded as the fourth difference of c.

These three methods of regarding the differences lead to three

different systems of notation, which are described in SS 9, 10 and 11.

_Notation of Differences and Sums._

8. It is convenient to denote the terms a, b, c, ... of the series by

u0, u1, u2, u3, ... If we merely have the terms of the series, un may

be regarded as meaning the (n + 1)th term. Usually, however, the terms

are the values of a quantity u, which is a function of another

quantity x, and the values of x, to which a, b, c, ... correspond,

proceed by a constant difference h. If x0 and u0 are a pair of

corresponding values of x and u, and if any other value x0 + mh of x

and the corresponding value of u are denoted by xm and um, then the

terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),

u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),

x_(n-1), x_n, x_(n+1), x_(n+2)....

9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by

[Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may

then be regarded as values of a function [Delta]u corresponding to

values of x proceeding by constant difference h; and therefore

[Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more

briefly, [Delta]^2u_n; and so on. Hence the table of differences in S

2, with the corresponding values of x and of u placed opposite each

other in the ordinary manner of mathematical tables, becomes

+---------+---------+----------------+-----------------+-----------------+----------------------+

+---------+---------+----------------+-----------------+-----------------+----------------------+

| . | . | . | . | . | . |

| | | | | | |

| x_(n-2) | u_(n-2) | | [Delta]^2u_(n-3)| | [Delta]^4u_(n-4) ... |

| x_(n-1) | u_(n-1) | | [Delta]^2u_(n-2)| | [Delta]^4u_(n-3) ... |

| xn | u_n | | [Delta]^2u_(n-1)| | [Delta]^4u_(n-2) ... |

| x_(n+1) | u_(n+1) | | [Delta]^2u_n | | [Delta]^4u_(n-1) ... |

| x_(n+2) | u_(n+2) | | [Delta]^2u_(n+1)| | [Delta]^4u_n ... |

| . | . | . | . | . | . |

+---------+---------+----------------+-----------------+-----------------+----------------------+

The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are

the first differences are denoted by [Sigma]u, with proper suffixes,

so that this series is ... [Sigma]u_(n-1), [Sigma]u_n,

[Sigma]u_(n+1).... The suffixes are chosen so that we may have

[Delta][Sigma]un = un, whatever n may be; and therefore (S 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]u_(n-1) = C + un-2,

where C is any constant, we shall have

[Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),

[Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,

and so on. This is true whatever C may be, so that the knowledge of

... u_n-1, u_n, ... gives us no knowledge of the exact value of

[Sigma]u_n; 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 u_(n+1), so that Eun = u_n +

[Delta]u_n.

10. Corresponding to the advancing-difference notation there is a

_receding-difference_ notation, in which u_(n+1) - u_n is regarded as

a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and

similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'^2u_(n+1).

This notation is only required for certain special purposes, and the

usage is not settled (S 19 (ii.)).

11. The _central-difference_ notation depends on treating u_(n+1) -

2u_n -u_(n-1) as the second difference of un, and therefore as

corresponding to the value x_n; but there is no settled system of

notation. The following seems to be the most convenient. Since un is a

function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is

a function of x_(n+1), the first difference u_(n+1) - u_n must be

regarded as a function of x_(n+1/2), i.e. of 1/2{x_n + x_(n+1)}. We

therefore write u_(n+1) - u_n = [delta]u_(n+1/2), and each difference in

the table in S 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]u_(n+1/2) = 1/2{u_n + u_(n+1)}, [mu][delta]u_n = 1/2{[delta]u_(n-1/2)} + [delta]u_(n+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:--

+-------+-------+---------------------+----------------+----------------------+----------------------+

+-------+-------+---------------------+----------------+----------------------+----------------------+

| . | . | . | . | . | . |

|x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}|[delta]^2u_(n-2)|{[mu][delta]^3u_(n-2)}| [delta]^4u_(n-2) ... |

|x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}|[delta]^2u_(n-1)|{[mu][delta]^3u_(n-1)}| [delta]^4u_(n-1) ... |

|x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}|[delta]^2u_(n+1)|{[mu][delta]^3u_(n+1)}| [delta]^4u_(n+1) ... |

|x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}|[delta]^2u_(n+2)|{[mu][delta]^3u_(n+2)}| [delta]^4u_(n+2) ... |

| . | . | . | . | . | . |

+-------+-------+---------------------+----------------+----------------------+----------------------+

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 x_(n-2) to x_(n-1), x_(n-1)

to x_n, ....

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]u_(n-3/2) = C + u_(n-2),

then

[sigma]u_(n-1/2) = C + u_(n-2) + u_(n-1), [sigma]_(n+1/2)

= C + u_(n-2) + u_(n-1) + u_n, &c.,

and also

[mu][sigma]u_(n-1) = C + u_(n-2) + 1/2u_(n-1), [mu][sigma]u_n

= C + u_(n-2) + u_(n-1) + 1/2u_n, &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]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an

_operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.

This applies to the other symbols E, [delta], &c., whether taken

simply or in combination. Thus [Delta]Eu_n means that we first replace

un by un+1, and then replace this by u_(n+2) - u_(n+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 (:=

d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =

[Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &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.u_n = u_(n+1) = u_n + [Delta]u_n = 1.u_n + [Delta].u_n,

so that we may write E = 1 + [Delta], or [Delta] = 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 E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and

expand by the binomial theorem and then operate on u0, we get the

general formulae

n.n - 1

un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,

1.2

n.n - 1

[Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,

1.2

which are identical with the formulae in (ii.) and (i.) of S 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]u_n = u_n, 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]v_r = u_r for

all integral values of r, then the sum of the terms u_m, u_(m+1), ...

un is v_(n+1) -v_m. 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 u_r is a rational integral

function of r of degree p, then [Delta]ur, 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

[Delta].(r+a+1) (r+a+2) ... (r+b) = (b-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-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 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

(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)

h^2

f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,

1.2

which, expressed in terms of operators, is

h^2 h^3

E = 1 + hD + ---D^2 + ----- D^3 + ... = e^(hD).

1.2 1.2.3

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