Concrete mathematics : a foundation for computer science

5.1 BASIC IDENTITIES 163
Two special cases of the binomial theorem are worth special attention,
even though they are extremely simple. If x = y = 1 and r = n is nonnegative,
we get
2n = (J+(y)+.-+(;), integer n 3 0.
This equation tells us that row n of Pascal’s triangle sums to 2”. And when
x is -1 instead of fl, we get
0" = (I)-(Y)+...+(-l)Q integer n 3 0.
For example, 1 - 4 + 6 - 4 + 1 = 0; the elements of row n sum to zero if we
give them alternating signs, except in the top row (when n = 0 and O” = 1).
When T is not a nonnegative integer, we most often use the binomial
theorem in the special case y = 1. Let’s state this special case explicitly,
writing z instead of x to emphasize the fact that an arbitrary complex number
can be involved here:
(1 +z)' = x (;)z*, IZI < 1.
k
(5.13)
The general formula in (5.12) follows from this one if we set z = x/y and
multiply both sides by y’.
We have proved the binomial theorem only when r is a nonnegative integer,
by using a combinatorial interpretation. We can’t deduce the general
case from the nonnegative-integer case by using the polynomial argument,
because the sum is infinite in the general case. But when T is arbitrary, we
can use Taylor series and the theory of complex variables:
f"(0)
+ FZ2 +...
The derivatives of the function f(z) = (1 + z)’ are easily evaluated; in fact,
fckl(z) = rk (1 + z)~~~. Setting 2 = 0 gives (5.13).
(Chapter 9 tells the We also need to prove that the infinite sum converges, when IzI < 1. It
meaning of 0 .) does, because (I) = O(k-‘-‘) by equation (5.83) below.
Now let’s look more closely at the values of (L) when n is a negative
integer. One way to approach these values is to use the addition law (5.8) to
fill in the entries that lie above the numbers in Table 155, thereby obtaining
Table 164. For example, we must have (i’) = 1, since (t) = (i’) + (11) and
(1:) = 0; then we must have (;‘) = -1, since (‘$ = (y’) + (i’); and so on.