Concrete mathematics : a foundation for computer science

As opposed to a
cassette function.
2.6 FINITE AND INFINITE CALCULUS 47
2.6 FINITE AND INFINITE CALCULUS
We’ve learned a variety of ways to deal with sums directly. Now it’s
time to acquire a broader perspective, by looking at the problem of summation
from a higher level. Mathematicians have developed a “finite calculus,”
analogous to the more traditional infinite calculus, by which it’s possible to
approach summation in a nice, systematic fashion.
Infinite calculus is based on the properties of the derivative operator D,
defined by
Df(x) = :rnO
f(x+ h) - f(x)
h ’
Finite calculus is based on the properties of the difference operator A, defined
by
Af(x) = f(x + 1) -f(x). (2.42)
This is the finite analog of the derivative in which we restrict ourselves to
positive integer values of h. Thus, h = 1 is the closest we can get to the
“limit” as h + 0, and Af(x) is the value of (f(x + h) - f(x))/h when h = 1.
The symbols D and A are called operators because they operate on
functions to give new functions; they are functions of functions that produce
functions. If f is a suitably smooth function of real numbers to real numbers,
then Df is also a function from reals to reals. And if f is any real-to-real
function, so is Af. The values of the functions Df and Af at a point x are
given by the definitions above.
Early on in calculus we learn how D operates on the powers f(x) = x"'.
In such cases Df(x) = mxmP’. We can write this informally with f omitted,
D(xm) = mx”-‘,
It would be nice if the A operator would produce an equally elegant result;
unfortunately it doesn’t. We have, for example,
A(x3) = (x+~)~-x’ = 3x 2 +3x+1.
Math power. But there is a type of “mth power” that does transform nicely under A,
and this is what makes finite calculus interesting. Such newfangled mth
powers are defined by the rule
m factors
A
XE = Ix(x-l)...(x-mmlj, integer m 3 0. (2.43)
Notice the little straight line under the m; this implies that the m factors
are supposed to go down and down, stepwise. There’s also a corresponding