(suggested new terminology)
I suggest ‘f(x) – f(red(x)’ as the (generalized) backward difference operator, where ‘red’ is short for ‘reduction of’. (This is a ‘horizontal’ generalization, as opposed to a ‘vertical’ generalization as might be needed in heavy-duty numerical analysis.) Two common forms of reduction of x are -x and x – 1. Actual use of the backward difference operator might occur in a kind of spread-out stepwise manner that disguises the fact that the backward difference operator is at work. For example, at a certain juncture in the derivation of the computational formula for the logarithm that Tom Apostol provides in volume 1 of his 2-volume Calculus textbook, there is an instance of the form ‘red(x) = -x’. A more directly visible occurrence of the backward difference operator is in specifying the general term in the summation of the average speed over members of a partition of an interval of time: (p(t(k)) – p(t(k – 1))/(t(k) – t(k – 1)), where p is the position function, the denominator of this expression being a ‘red(x) = x – 1’ instance of the backward difference operator. Another occurrence of the backward difference operator where ‘red(x)’ has the form ‘x – 1’ is the backward difference operator f(n) – f(n – 1) defined by Michael Z. Spivey in his article ‘The Humble Sum of Remainders Function’. (The Farlex Dictionary offers a generalized version as f(x) – f(x – h) – “where h is a constant denoting the difference between successive points of interpolation or calculation” – others calling h the ‘step size’.)
Another occurrence of the backward difference operator is in defining the Kaprekar process. The definition of the Kaprekar process can be worded as follows: “For a given 4-digit positive integer n (expressed in base 10, leading 0’s allowed) containing at least 2 digits, let f(n) be the largest number expressible using the digits of n, and let f(-n) be the smallest number expressible using the digits of n, and let k(1) = f(n) – f(-n). Then apply the preceding to k(1) as the ‘new n’, obtaining k(2), and so on.” (This process reaches the number 6174 in at most 7 steps.) Another occurrence of the backward difference operator is the probability formula P(A) = 1 – P(A’),
which can be written as P(A) = P(S) – P(red(S)), where red(S) = S – A. In a similar way, the expression 1 – 1/p, which can be written as i(p/p) – i(red(p/p), where red(p/p) = 1/p (i being the identity function) can be considered an occurrence of the the backward difference operator, which Euler put to good use in evaluating ζ(2). Another occurrence of the backward difference operator is the definition of the interquartile range (IQR), namely, Q(3) – Q(1), which can be written in the form Q(3) – Q(red(3)), where red(3) = 1.
Just as group theory can be use additive notation (typically, for abelian groups) or multiplicative notation, so also the ‘red(x) = -x’ version of the backward difference operator can be stated in multiplicative form as well: f(x)/(f(1/x).
[backward difference formula]
[f(x) – f(-x)]
[f(x) minus f(-x)]
[f(x) minus f(reduction of(x))]
[f(x) minus f(red(x))]
(suggested new terminology)