Differences, Calculus Of

Differences, Calculus Of (Theory of Finite Differences), that branch of mathematics which deals with the successive differences of the terms of a series.

1. The most important of the cases to which mathematical methods can be applied are those in which the terms of the series are the values, taken at stated intervals (regular or irregular), of a continuously varying quantity. In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e. the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry), i.e. where the value of each term depends on the values of a pair of other quantities.

2. The first differences of a series are obtained by subtracting from each term the term immediately preceding it. If these are treated as terms of a new series, the first differences of this series are the second differences of the original series; and so on. The successive differences are also called differences of the first, second, ... order. The differences of successive orders are most conveniently arranged in successive columns of a table thus:—

Term.	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
a
	b − a
b		c − 2b + a
	c − b		d − 3c + 3b − a
c		d − 2c + b		e − 4d + 6c − 4b + a
	d − c		e − 3d + 3c − b
d		e − 2d + c
	e − d
e

Algebra of Differences and Sums.

Fig. 1.

3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms α, β, γ, δ, commencing with the same term α, and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law; i.e. the coefficients in b − a, c − 2b + a, d − 3c + 3b − a, ... and in α + β, α + 2β + γ, α + 3β + 3γ + δ, ... are respectively the same as in y − x, (y − x)², (y − x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B); e.g. β + γ is the difference of α + 2β + γ and α + β, just as c − b is the difference of c and b: and d − c is the sum of c − b and d − 2c + b, just as β + 2γ + δ is the sum of β + γ and γ + δ. Hence if we take β, γ, δ, ... of (B) as being the same as b − a, c − 2b + a, d − 3c + 3b − a, ... of (A), all corresponding terms in the two diagrams will be the same.

Thus we obtain the two principal formulae connecting terms and differences. If we provisionally describe b − a, c − 2b + a, ... as the first, second, ... differences of the particular term a (§ 7), then (i.) the nth difference of a is

l − nk + ... + (-1)n-2	n·n − 1	c + (-1)n-1 nb + (-1)n a,
	1·2

where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c, ...; the coefficients being those of the terms in the expansion of (y − x)n: and (ii.) the (n + 1)th term of the series, i.e. the nth term after a, is

a + nβ +	n·n − 1	γ + ...
	1·2

where β, γ, ... are the first, second, ... differences of a; the coefficients being those of the terms in the expansion of (x + y)n.

4. Now suppose we treat the terms a, b, c, ... as being themselves the first differences of another series. Then, if the first term of this series is N, the subsequent terms are N + a, N + a + b, N + a + b + c, ...; i.e. the difference between the (n + 1)th term and the first term is the sum of the first n terms of the original series. The term N, in the diagram (A), will come above and to the left of a; and we see, by (ii.) of § 3, that the sum of the first n terms of the original series is

(	N + na +	n·n − 1	β + ...	)	− N = na +	n·n − 1	β +	n·n − 1·n − 2	γ + ...
		1·2				1·2		1 · 2 · 3

5. As an example, take the arithmetical series

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 § 3, the (n + 1)th term is a + np, and, by § 4, the sum of the first n terms is na + ½n(n − 1)p = ½n{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 (§ 14 (i.)) that all differences of a higher order are zero. Hence the sum of the first n terms is

n + 7	n·n − 1	+ 12	n·n − 1·n − 2	+ 6	n·n − 1·n − 2·n − 3	= ¼n4 + ½n³ + ¼n² = {½n (n + 1)}².
	1·2		1·2·3		1·2·3·4

7. In § 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 § 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 §§ 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 ... un-2, un-1, un, un+1, un+2 ..., corresponding to values of x denoted by ... xn-2, xn-1, xn, xn+1, xn+2....

9. In the advancing-difference notation un+1 − un is denoted by Δun. The differences Δu0, Δu1, Δu2 ... may then be regarded as values of a function Δu corresponding to values of x proceeding by constant difference h; and therefore Δun+1 − Δun denoted by ΔΔun, or, more briefly, Δ²un; and so on. Hence the table of differences in § 2, with the corresponding values of x and of u placed opposite each other in the ordinary manner of mathematical tables, becomes

x	u	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·
xn-2	un-2		Δ²un-3		Δ4un-4 ...
		Δun-2		Δ³un-3
xn-1	un-1		Δ²un-2		Δ4un-3 ...
		Δun-1		Δ³un-2
xn	un		Δ²un-1		Δ4un-2 ...
		Δun		Δ³un-1
xn+1	un+1		Δ²un		Δ4un-1 ...
		Δun+1		Δ³un
xn+2	un+2		Δ²un+1		Δ4un  ...
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·

The terms of the series of which ... un-1, un, un+1, ... are the first differences are denoted by Σu, with proper suffixes, so that this series is ... Σun-1, Σun, Σun+1.... The suffixes are chosen so that we may have ΔΣun = un, whatever n may be; and therefore (§ 4) Σun may be regarded as being the sum of the terms of the series up to and including un-1. Thus if we write Σun-1 = C + un-2, where C is any constant, we shall have

Σun = Σun-1 + ΔΣun-1 = C + un-2 + un-1,

Σun+1 = C + un-2 + un-1 + un,

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 Σun; 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 Σu 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 + Δun.

10. Corresponding to the advancing-difference notation there is a receding-difference notation, in which un+1 − un is regarded as a difference of un+1, and may be denoted by Δ′un+1, and similarly un+1 − 2un + un-1 may be denoted by Δ′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 − 2un − 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 − 2un+1 + un is a function of xn+1, the first difference un+1 − un must be regarded as a function of xn+1/2, i.e. of ½(xn + xn+1). We therefore write un+1 − un = δun+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 μ 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

μun+1/2 = ½(un + un+1, μδun = ½(δun-1/2 + δun+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	u	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·
xn-2	un-2	(μδun-2)	δ²un-2	(μδ³un-2)	δ4un-2 ...
		δun-3/2		δ³un-3/2
xn-1	un-1	(μδun-1)	δ²un-1	(μδ³un-1)	δ4un-1 ...
		δun-1/2		δ³un-2
xn	un	(μδun)	δ²un	(μδ³un)	δ4un  ...
		δun+1/2		δ³un+1/2
xn+1	un+1	(μδun+1)	δ²un+1	(μδ³un+1)	δ4un+1 ...
		δun+3/2		δ³un+3/2
xn+2	un+2	(μδun+2)	δ²un+2	(μδ³un+2)	δ4un+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 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 σu, with suffixes on the same principle; the suffixes being chosen so that δσun shall be equal to un. Thus, if

σun-3/2 = C + un-2,

then

σun-1/2 = C + un-2 + un-1, σn+1/2 = C + un-2 + un-1 + un, &c.,

and also

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

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).

(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 Δvn = un+1, which may also be written (E − 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 − p1)(E − p2) ... (E − 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.) 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 ∂u0, ∂²u0 ..., we have (§ 3 (ii.))

u1 = u0 + q∂u0 +	q·q − 1	∂²u0 + ....
	1·2

When q is made indefinitely great, this (writing ƒ(x) for u) becomes Taylor’s Theorem (Infinitesimal Calculus)

ƒ(x + h) = ƒ(x) + hƒ′(x) +	h²	ƒ″(x) + ...,
	1·2

which, expressed in terms of operators, is

E = 1 + hD +	h²	D² +	h³	D³ + ... = ehD.
	1·2		1·2·3

This gives the relation between Δ and D. Also we have

u2 = u0 + 2q∂u0 +	2q·2q − 1	∂²u0 + ...
	1·2
u3 = u0 + 3q∂u0 +	3q·3q − 1	∂²u0 + ...
	1·2
·   ·
·   ·
·   ·

and, if p is any integer,

up/q = u0 + p∂u0 +	p·p − 1	∂²u0 + ....
	1·2

From these equations up/q could be expressed in terms of u0, u1, u2, ...; this is a particular case of interpolation (q.v.).

18. Differences and Differential Coefficients.—The various formulae are most quickly obtained by symbolical methods; i.e. by dealing with the operators Δ, E, D, ... as if they were algebraical quantities. Thus the relation E = ehD (§ 17) gives

hD = loge (1 + Δ) = Δ − ½Δ² + 1⁄3Δ³ ...

or

h(du/dx)0 = Δu0 − ½Δ²u0 + 1⁄3Δ³u0 ....

The formulae connecting central differences with differential coefficients are based on the relations μ = cosh ½hD = ½(e1/2hD + e-1/2hD), δ = 2 sinh ½hD − e1/2hD − e-1/2hD, and may be grouped as follows:—

u0	= u0	}
μδu0	= (hD + 1⁄6 h³D³ + 1⁄120 h5D5 + ...)u0
δ²u0	= (h²D² + 1⁄12 h4D4 + 1⁄360 h6D6 + ...)u0
μδ³u0	= (h³D³ + 1⁄4 h5D5 + ...)u0
δ4u0	= (h4D4 + 1⁄6 h6D6 + ...)u0
	·   ·   ·
	·   ·   ·
	·   ·   ·
μu1/2	= (1 + 1⁄8 h²D² + 1⁄384 h4D4 + 1⁄46080 h6D6 + ...)u1/2	}
δu1/2	= (hD + 1⁄24 h³D³ + 1⁄1920 h5D5 + ...)u1/2
μδ²u1/2	= (h²D² + 5⁄24 h4D4 + 91⁄5760 h6D6 + ...)u1/2
δ³u1/2	= (h³D³ + 1⁄8 h5D5 + ...)u1/2
μδ4 u1/2	= (h4D4 + 7⁄24 h6D6 + ...)u1/2
	·   ·   ·
	·   ·   ·
	·   ·   ·
u0	= u0	}
hDu0	= (μδ − 1⁄6 μδ³ + 1⁄30 μδ5 − ...)u0
h²D²u0	= (δ² − 1⁄12 δ4 + 1⁄90 δ6 − ...)u0
h³D³u0	= (μδ³ − 1⁄4 μδ5 + ...)u0
h4D4u0	= (δ4 − 1⁄6 δ6 + ...)u0
	·   ·   ·
	·   ·   ·
	·   ·   ·
u1/2	= (μ − 1⁄8 μδ² + 3⁄128 μδ4 − 5⁄1024 μδ6 + ...)u1/2	}
hDu1/2	= (δ − 1⁄24 δ³ + 3⁄640 δ5 − ...)u1/2
h²D²u1/2	= (μδ² − 5⁄24 μδ4 + 259⁄5760 μδ6 − ...)u1/2
h³D³u1/2	= (δ³ − 1⁄8 δ5 + ...)u1/2
h4D4 u1/2	= (μδ4 − 7⁄24 μδ6 + ...)u1/2
	·   ·   ·
	·   ·   ·
	·   ·   ·

When u is a rational integral function of x, each of the above series is a terminating series. In other cases the series will be an infinite one, and may be divergent; but it may be used for purposes of approximation up to a certain point, and there will be a “remainder,” the limits of whose magnitude will be determinate.

19. Sums and Integrals.—The relation between a sum and an integral is usually expressed by the Euler-Maclaurin formula. The principle of this formula is that, if um and um+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between um and um+1 we have ½h(um + um+1), and the difference between this and the true value of the area can be expressed as the difference of two expressions, one of which is a function of xm, and the other is the same function of xm+1. Denoting these by φ(xm) and φ(xm+1), we have

∫	xm+1	udx = ½h(um + um+1) + φ(xm+1) − φ(xm).
	xm

Adding a series of similar expressions, we find

∫	xn	udx = h{½um + um+1 + um+2 + ... + un-1 + ½un} + φ(xn) − φ(xm).
	xm

The function φ(x) can be expressed in terms either of differential coefficients of u or of advancing or central differences; thus there are three formulae.

(i.) The Euler-Maclaurin formula, properly so called, (due independently to Euler and Maclaurin) is

∫

xn

udx = h·μσun − 1⁄12 h²

dun

½ + 1⁄720 h4

d³un

− 1⁄30240 h6

d5un

+ ... = h·μσun −

B1

h2

dun

+

B2

h4

d3un

−

B3

h6

d5un

+ ...

dx

dx3

dx5

2!

dx

4!

dx3

6!

dx5

where B1, B2, B3 ... are Bernoulli’s numbers.

(ii.) If we express differential coefficients in terms of advancing differences, we get a theorem which is due to Laplace:—

1/h	∫	xn	udx = μσ(un − u0) − 1⁄12(Δun − Δu0) + 1⁄24(Δ²un − Δ²u0) − 19⁄720(Δ³un − Δ³u0) + 3⁄160(Δ4un − Δ4u0) − ...
		x0

For practical calculations this may more conveniently be written

1/h	∫	xn	udx = μσ(un − u0) + 1⁄12(Δu0 − ½Δ²u0 + 19⁄60Δ³u0 − ...) + 1⁄12(Δ′ un − ½Δ′ ²un + 19⁄60Δ′ ³un − ...),
		x0

where accented differences denote that the values of u are read backwards from un; i.e. Δ′un denotes un-1 − un, not (as in § 10) un − un-1.

(iii.) Expressed in terms of central differences this becomes

1/h	∫	xn	udx = μσ(un − u0) − 1⁄12μδun + 11⁄720 μδ³un − ... + 1⁄12μδu0 − 11⁄720 μδ³u0 + ... = μ(σ − 1⁄12δ + 11⁄720δ³ − 191⁄60480δ5 + 2497⁄3628800δ7 − ...)(un − u0).
		x0

(iv.) There are variants of these formulae, due to taking hum+1/2 as the first approximation to the area of the curve between um and um+1; the formulae involve the sum u1/2 + u3/2 + ... + un-1/2 ≡ σ(un − u0) (see Mensuration).

20. The formulae in the last section can be obtained by symbolical methods from the relation

1/h

∫

udx = 1/h D-1u = 1/hD · u.

Thus for central differences, if we write θ ≡ ½hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ-1, and the result in (iii.) corresponds to the formula

sinh θ = θ cosh θ/(1 + 1⁄3 sinh² θ − 2⁄3·5 sinh4 θ + 2·4⁄3·5·7 sinh6 θ − ...).

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