Page 47 - Foundations Of Differential Calculus

30 2. On the Use of Differences in the Theory of Series
46. The differences, as we have seen above, can be expressed in terms of
the original series as follows:
I II I III II I
b = a − a, c = a − 2a + a, d = a − 3a +3a − a,
I II I I III II I I IV III II I
b = a − a , c = a − 2a + a , d = a − 3a +3a − a ,
II III II II IV III II II V IV III II
b = a − a ,c = a − 2a + a ,d = a − 3a +3a − a ,
..., ..., ....

Since in a series of the first order all values of c vanish, we have
II I III II I IV III II
a =2a − a, a =2a − a , a =2a − a , ...,

and so it is clear that these series are recurrent and that the scale of relation
is 2, −1. Then, since in a series of the second order all values of d vanish,
we have
III II I IV III II I
a =3a − 3a + a, a =3a − 3a + a , ....
From this it follows that these series are recurrent with a scale of relation
3, −3, 1. In a similar way it can be shown that each such series of any order
is both a recurrent series and the scale of relation consists of the binomial
coefficients where the exponent is one more than the order of the series.
47. Since in a series of the first order we also have all d’s, e’s, and all of
the subsequent differences vanishing, we also have
III II I
a =3a − 3a + a,
I
II
a IV =3a III − 3a + a ,
and so forth, or
IV III II I
a =4a − 6a +4a − a,
V IV III II I
a =4a − 6a +4a − a ,
and so forth. From this it follows that these are recurrent series, indeed in
an infinite number of ways, since the scales of relation can be 3, −3, +1;
4, −6, +4, −1; 5, −10, +10, −5+1; ... . In a similar way it should be un-
derstood that each of the series of the kind we have been discussing is a
recurrent series in an infinite number of ways and that the scale of relation
is
n (n − 1) n (n − 1) (n − 2) n (n − 1) (n − 2) (n − 3)
n, − , + , − ,
1 · 2 1 · 2 · 3 1 · 2 · 3 · 4