Concrete mathematics : a foundation for computer science

7.3 SOLVING RECURRENCES
7.3 SOLVING RECURRENCES 323
Now let’s focus our attention on one of the most important uses of
generating functiorrs: the solution of recurrence relations.
Given a sequence (gn) that satisfies a given recurrence, we seek a closed
form for gn in terms of n. A solution to this problem via generating functions
proceeds in four steps that are almost mechanical enough to be programmed
on a computer:
1 Write down a single equation that expresses g,, in terms of other elements
of the sequence. This equation should be valid for all integers n, assuming
that g-1 = g-2 = ... = 0.
2 Multiply both sides of the equation by zn and sum over all n. This gives,
on the left, the sum x., gnzn, which is the generating function G (2). The
right-hand side should be manipulated so that it becomes some other
expression involving G (2).
3 Solve the resulting equation, getting a closed form for G (2).
4 Expand G(z) into a power series and read off the coefficient of zn; this is
a closed form for gn.
This method works because the single function G(z) represents the entire
sequence (gn) in such a way that many manipulations are possible.
Example 1: Fibonacci numbers revisited.
For example, let’s rerun the derivation of Fibonacci numbers from Chapter
6. In that chapter we were feeling our way, learning a new method; now
we can be more systematic. The given recurrence is
go = 0; 91 = 1;
gn = %-1+%-z, for n 3 2.
We will find a closed form for g,, by using the four steps above.
Step 1 tells us to write the recurrence as a “single equation” for gn. We
could say
9 n=
0, ifn 1;
but this is cheating. Step 1 really asks for a formula that doesn’t involve a
case-by-case construction. The single equation
gn = gn-l+~ln-z
works for n > 2, a.nd it also holds when n 6 0 (because we have go = 0
and gnegative = 0). But when n = 1 we get 1 on the left and 0 on the right.