Concrete mathematics : a foundation for computer science

I always thought
convolution was
what happens to
my brain when 1
try to do a proof.
7.4 SPECIAL GENERATING FUNCTIONS 339
Thus, for example, the case n = 1 of (7.51) should not be regarded as the
power series ,&,O(zn/n!){, l,}/(z), but rather as
z
ln(1 + 2)
= -t(-z)“oll(n-l) = 1 +~z-~zz+... .
II20
Identities (7.53), (7.551, (7.54), and (7.56) are “double generating functions”
or “super generating functions” because they have the form G (w, z) =
t,,, Sm,n~“‘2~. The coefficient of wm is a generating function in the variable
z; the coefficient of 2” is a generating function in the variable w.
7.5 CONVOLUTIONS
The convolution of two given sequences (fo, fl , . . ) = (f,,) and
(SOlSl,. . .) = (gn) is the sequence (f0g0, fog1 + flg0, . . .) = (xkfkgn k).
We have observed in Sections 5.4 and 7.2 that convolution of sequences corresponds
to multiplication of their generating functions. This fact makes it
easy to evaluate many sums that would otherwise be difficult to handle.
Example 1: A Fibonacci convolution.
For example, let’s try to evaluate ~~=, FkFn~-k in closed form. This is
the convolution of (F,) with itself, so the sum must be the coefficient of 2”
in F(z)', where F(z) is the generating function for (F,). All we have to do is
figure out the value of this coefficient.
The generating function F(z) is z/( 1 -z-z’), a quotient of polynomials; so
the general expansion theorem for rational functions tells us that the answer
can be obtained from a partial fraction representation. We can use the general
expansion theorem (7.30) and grind away; or we can use the fact that
Instead of expressing the answer in terms of C$ and $i, let’s try for a closed
form in terms of Fibonacci numbers. Recalling that Q + $ = 1, we have
$“+$” = [z”l j& + &J
( 2- (Q+$)z
= Lz"' (1 - ($z)( 1 _ qjz) = VI
2-z
l-Z-22
= 2F,+, -F,.