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CHAPTER II.

DIRECT THEOREMS OF FINITE DIFFERENCES.

1. THE operation denoted by A is capable of repetition. For the difference of a function of x, being itself a function of x, is subject to operations of the same kind.

In accordance with the algebraic notation of indices, the difference of the difference of a function of x, usually called the second difference, is expressed by attaching the index 2 to the symbol A. Thus

the last member being termed the nth difference of the function
Ux
If we suppose u= x3, the successive values of
Их
with
their successive differences of the first, second, and third orders
will be represented in the following scheme:

It may be observed that each set of differences may either be formed from the preceding set by successive subtractions in accordance with the definition of the symbol ▲, or calculated from the general expressions for Au, A3u, &c. by assign

ing to x the successive values 1, 2, 3, &c. Since u2 = shall have

It may also be noted that the third differences are here constant. And generally if u be a rational and integral function of x of the nth degree, its nth differences will be constant. For let

b1, b, &c., being constant coefficients. Hence Au is a rational and integral function of x of the degree n-1. Repeating the process, we have

2-2

n-4

▲3u2 = an (n − 1) 1⁄2”−2 +¢ ̧1⁄2” ̄3 + С2xˆ ̄ + &c.,

a rational and integral function of the degree n-2; and so on.

2. While the operation or series of operations denoted by A, A3, ... A" are always possible when the subject-function u is given, there are certain elementary cases in which the forms of the results are deserving of particular attention, and these we shall next consider.

Differences of Elementary Functions.

1st. Let u2 = x (x − 1) (x − 2) ... (x − m + 1).

Then by definition,

▲u ̧=(x+1)x(x−1)...(x−m+2)−x (x−1)(x-2)... (x−m+1)

= mx(x-1) (x-2)... (x — m + 2).

When the factors of a continued product increase or decrease by a constant difference, or when they are similar functions of a variable which, in passing from one to the other, increases or decreases by a constant difference, as in the expression

sin x sin (x + h) sin (x + 2h) ... sin {x + (m − 1) h},

the factors are usually called factorials, and the term in which they are involved is called a factorial term. For the particular kind of factorials illustrated in the above example it is common to employ the notation

Hence, am-1) being also a factorial term,

▲3xm) = m (m −1) x(m−2),

'= m (m − 1) ... (m − n + 1) xm

and generally

Then by definition,

Aux

=

(x+1)(x+2)... (x+m)

1

1

x (x + 1) ... (x + m − 1)

-) (x + 1) (x + 2) ... (∞ + m − 1)

m

x(x+1)... (x+m)

Hence by successive repetitions of the operation ▲,

▲"x( ̄m) — — m ( — m − 1) ... (— m − n + 1) x(−m-n)

=

-

=(−1)" m (m + 1) ... (m + n − 1) x ̄m-n)

and this may be regarded as an extension of (3).

3rdly. Employing the most general form of factorials,

we find

4thly. To find the successive differences of a".

5thly. To deduce the successive differences of sin (ax + b) and cos (ax+b).

▲ sin (ax + b) = sin (ax + b + a) − sin (ax+b)

By inspection of the form of this result we see that

▲3 sin (ax + b) = (2 sin 2)* sin (ax+b+a+π)...........(16).

These results might also be deduced by substituting for the sines and cosines their exponential values and applying (15).

3. The above are the most important forms. The following are added merely for the sake of exercise.