Concrete mathematics : a foundation for computer science

52 SUMS
we need an appropriate definition of ~3 for m < 0. Looking at the sequence
x3 = x(x-1)(x-2),
XL = x(x-l),
x1 = x,
XQ = 1,
we notice that to get from x2 to x2 to xl to x0 we divide by x - 2, then
by x - 1, then by X. It seems reasonable (if not imperative) that we should
divide by x + 1 next, to get from x0 to x5, thereby making x5 = 1 /(x + 1).
Continuing, the first few negative-exponent falling powers are
1
x;1 = - x+1 '
x-2 = (x+*:(x+2) '
1
x-3 = (x+1)(x+2)(x+3)
and our general definition for negative falling powers is
'-"' = (x+l)(x+2)...(x+m)
1
for m > 0. (2.51)
(It’s also possible to define falling powers for real or even complex m, but we How can a complex
will defer that until Chapter 5.)
number be even?
With this definition, falling powers have additional nice properties. Perhaps
the most important is a general law of exponents, analogous to the law
Xm+n = XmXn
for ordinary powers. The falling-power version is
xmi-n = xZ(x-m,)n, integers m and n.
For example, xs = x1 (x - 2)z; and with a negative n we have
x23 zz xqx-q-3 = x(x- 1)
1
= -
1
= x;l,
(x- 1)x(x+ 1) x+1
(2.52)
If we had chosen to define xd as l/x instead of as 1 /(x + l), the law of
exponents (2.52) would have failed in cases like m = -1 and n = 1. In fact,
we could have used (2.52) to tell us exactly how falling powers ought to be
defined in the case of negative exponents, by setting m = -n. When an Laws have their
existing notation is being extended to cover more cases, it’s always best to exponents and their
formulate definitions in such. a way that general laws continue to hold.
detractors.