Algebra of Differences and Sums

ALGEBRA OF DIFFERENCES AND SUMS 2. Let p, q, r, s, t • • • be consecutive terms of a series. Then by successive subtractions we get the first differences q— p, r—q • • • , the second differences r— s— 2r- -1- • • • , and so on. These are also called differences of the first, second • • • , order. It is convenient to arrange the terms in one column, and the successive differences in successive columns, as in Table I.

TABLE I.Table i.

Term. ist dill. 2nd diff. 3rd diff. 4th dill.

p q r— 2q-k p s-3r+3q---p ? s-2r-+-q t-4s+6r-4q+p s—r —q S t-2s-Fr t—s It will be noticed that the coefficients in the successive columns of differences are those of the powers of y in y— I, (y (y— • • • . For considering relations of this kind we need a special notation.

3. Systems of Notation.

We shall denote the terms of the series by • • • • • • , the suffixes o, I, 2 • • • merely indicating the position of the term in the series. If the series begins with a definite term, we usually call this or according to circumstances. But in a good many cases there is no first term, the series being capable of contipuation backwards as well as forwards. Thus in the example given in § i (i.) we could introduce the squares of negative integers as well as those of positive in tegers, and the series would then be:— No. . . . —I o I 2 3 • • Square . • 4 I o I 4 9••• Ist diff. . . I 3 5 so that the differences would be the odd numbers, negative and positive.

The difference of two consecutive u's can then be denoted by Liu; the difference of two consecutive differences by &u, or, for shortness, and so on. But we again need a system of suffixes to indicate where a particular difference comes. Now, in §2, the expression q— p contains p and q, and it might be re garded as the first difference either of p or of q. On the one system the quantity 1-4s+6r-4q+P would be the fourth difference of p; on the other system it would be the fourth difference of t. Or, since the term which has the greatest coefficient in this quan tity is r, the quantity might be regarded as the fourth difference of r. These three possible ways of regarding the matter lead to three different systems of notation, which are called the advancing difference, the receding-difference, and the central-difference sys tems, respectively. We need only consider the first and the third of these.

4. Advancing-difference Notation.

In the advancing-dif ference notation we regard as the first difference of and give to this value of Au the suffix n, i.e., we have 5. Separation of Symbols of Operation.—We can regard as the result of a certain operation performed on u,,, just as 5A denotes the result of the operation of multiplying A by 5, i.e., of changing A into 5 times A. The operation of changing to consists of two steps : (a) changing to (b) sub tracting from the result. We use the symbol E to denote the first of these two steps, i.e., • We then have (3) The important property of these symbols is that, like numerical coefficients, they can be detached from the term on which they operate and can be dealt with according to the laws of ordinary algebra, as if they were numbers. Thus, if we wish we may write (3) as follows: =E—i, E=I+O. (4) As an example of the application of the property mentioned above, we have Onuo = I)nuo = )En . . . uo (5) Ii n(n--I) = un-2 • • • I This is the relation mentioned at the end of §2. Similarly we find that = i -}-nQ -}-n (n-1) . . . +Al (6) = uo (n— . This relation is of importance for the purpose of finding the sums of certain series (see §q).

6. taking the table in §4, let us continue it backwards, i.e., let us regard the u's as the first differences of the terms of another series, which we will call • • • `atn+l • • • . Then for consistency of notation we see that must come be tween and i.e., , 2{n_}-1 , • • . (7) Thus we get Table III. Similarly, we can insert further columns containing etc.

8. Operation of Taking the Mean.—We introduce the symbol ΅ to denote the operation of taking the mean of two ad jacent quantities in a column, the suffix, like that of 6, being the mean of their suffixes. Thus yun+1 ° 2 (un+un+l), 2 etc. These means may be called the constructed central differences. If they are introduced, in brackets, into Table IV., we get the complete central-difference table of which Table V. is a part.

TABLE V.Table v.

u 6u 62, If we only know the u's, we cannot find the new series com pletely; for the accuracy of the above table is not affected by adding a constant k to each term of the /u series. But, when we have found a particular term of this series, say /u,, the other terms are also fixed. By (7) we then have Zu3= ZUl+ul+u2 • • • (8) /u,+ul+u2+ • • • -Fun-1 and also A I = I I un = un. (9) The relation between the operators A and / may be written in either of the forms I, (to) It should be observed that does not include it only goes as far as u,,_,.

Until we know a term of the /u series, we must leave the actual terms indefinite, their first differences being definite. Thus we can write Eun= C+un-3+un^2+un-1 where C is an "arbitrary constant"; and we shall then have, with the same value of C, L /u„.0= C+un-3+un-2+un-l+un, and so on.

7. Central-difference Notation.—The central-difference no tation differs from the advancing-difference in two respects: (i.) the assignment of suffixes, (ii.) the introduction of a symbol to denote the operation of taking the mean.

The main principle, as indicated in §3, is that un+2— 2un+i+un is regarded as the second difference not of but of it is denoted by The first difference — is therefore 6u, , with a proper suffix; and this suffix, for symmetry, must be n+1. Thus we have 6 un+} = un+l — = — etc. (II) On the same principle the entries in the column preceding the u column will be au, with proper suffixes; and we have = — au,_1, etc., so that the operations 6 and a are connected by the relation &T= I, (I2) Our table, from the column to the column, becomes Table IV.

It is obvious that au„+} = C+un-3+un-2+2 -l+un 1 / \i = C+un-3+un-2+un-1+ZUn 4) where C is an arbitrary constant which remains the same through out any series of operations.