Concrete mathematics : a foundation for computer science

48 SUMS
definition where the factors go up and up:
m factors
I h .
x iii = x(x+l)...(x+m-l), integer m 3 0. (2.44)
When m = 0, we have XQ = x-’ = 1, because a product of no factors is
conventionally taken to be 1 (just as a sum of no terms is conventionally 0).
The quantity xm is called “x to the m falling,” if we have to read it
aloud; similarly, xK is “x to the m rising!’ These functions are also called
falling factorial powers and rising factorial powers, since they are closely
related to the factorial function n! = n(n - 1). . . (1). In fact, n! = nz = 1”.
Several other notations for factorial powers appear in the mathematical
literature, notably “Pochhammer’s symbol” (x), for xK or xm; notations
like xc”‘) or xlml are also seen for x3. But the underline/overline convention
is catching on, because it’s easy to write, easy to remember, and free of
redundant parentheses.
Falling powers xm are especially nice with respect to A. We have
A(G) = (x+1)=-x”
= (x+1)x.. .(x-m++) - x... (x--+2)(x-m+l)
= mx(x-l)...(x-m+2),
hence the finite calculus has a handy law to match D(x”‘) = mx”-‘:
A(x”) = mxd. (2.45)
This is the basic factorial fact.
The operator D of infinite calculus has an inverse, the anti-derivative
(or integration) operator J. The Fundamental Theorem of Calculus relates D
to J:
g(x) = Df(xl if and only if g(x) dx = f(x) + C.
Here s g(x) dx, the indefinite integral of g(x), is the class of functions whose
derivative is g(x). Analogously, A has as an inverse, the anti-difference (or
summation) operator x; and there’s another Fundamental Theorem:
g(x) = Af(xl if and only if xg(x)bx = f(x)+C. (2.46)
Here x g(x) 6x, the indefinite sum of g(x), is the class of functions whose
diflerence is g(x). (Notice that the lowercase 6 relates to uppercase A as
d relates to D.) The “C” for indefinite integrals is an arbitrary constant; the
“C” for indefinite sums is any function p(x) such that p(x + 1) = p(x). For
Mathematical
terminology is
sometimes crazy:
Pochhammer 12341
actually used the
notation (x) m
for the binomial
coefficient (k) , not
for factorial powers.
“Quemadmodum
ad differentiam
denotandam usi
sumus sign0 A,
ita summam indicabimus
sign0 L.
. . . ex quo zquatio
z = Ay, siinvertatur,
dabit quoque
y = iEz+C.”
-L. Euler /88]