In the difference table for ln x, the first column corresponded to equi-distant data - values of f(x):
The following shift operator E is defined by
where k need not be positive or an integer:
Why should we be concerned with fractional powers of E? To start with, we might only want to use this formula with every second value in a table and later switch over to every value.
In the difference table above, we formed the first differences fj + fj -1 , which in terms of the operator E are
since
Similarly,
In this manner, binomial coefficients enter the finite difference calculus, since
The central difference operator is defined by:
the backward operator by:
The corresponding differences tables are
The values of the forward and backward differences are the same, but they occupy different positions in the tables. The odd order central differences have other values, the values of the even differences agree with those of the other differences. This aspect of finite differences becomes important for interpolation. In order to interpolate function values at the beginning of a table, forward differences must used, because they have the largest number of differences, at the centre of their table, central differences are used inside the table, and backward differences near the ends. This becomes obvious when one rewrites the tables so that the differences of different orders with the same subscripts are written on one line. Then the front of the forward difference table slants backwards, that of the backward table forward, while the central difference table has triangular form:
