Concrete mathematics : a foundation for computer science

7.2 BASIC MANEUVERS 319
Replacing the z by a constant multiple is another of our tricks:
G(u) = t ~,(cz)~ = xcngnz”; (7.16)
n n
this yields the generating function for the sequence (c”g,). The special case
c = -1 is particularly useful.
I fear d genera ting- Often we want to bring down a factor of n into the coefficient. Differenfunction
dz 3. tiation is what lets us ‘do that:
G’(z) = gl +2g2z+3g3z2+- = t(n+l)g,+,z". (7.17)
n
Shifting this right one place gives us a form that’s sometimes more useful,
zG’(z) = tng,,z”
n
(7.18)
This is the generating function for the sequence (ng,). Repeated differentiation
would allow us to multiply g,, by any desired polynomial in n.
Integration, the inverse operation, lets us divide the terms by n:
J L G(t)dt = gez+ fg,z2 + ;g2z3 +... = x 1 p-d. (7.19)
0 TI>l
(Notice that the constant term is zero.) If we want the generating function
for (g,/n) instead of (g+l/n), we should first shift left one place, replacing
G(t) by (G(t) - gc)/t in the integral.
Finally, here’s how we multiply generating functions together:
F(z)G(z) =
=
(fo+f,z+f2z2+~-)(go+g1z+g2z2+-~)
(fogo) + (fog1 +f1!Ilo)z + (fog2 +f1g1 +f2go)z2 + ...
~(-pk&k)ZTI. (7.20)
TL k
As we observed in Chapter 5, this gives the generating function for the sequence
(hn), the convolution of (fn) and (gn). The sum hn = tk fk&-k can
also be written h, = ~~=, fkgnpkr because fk = 0 when k < 0 and gn-k = 0
when k > n. Multiplication/convolution is a little more complicated than
the other operations, but it’s very useful-so useful that we will spend all of
Section 7.5 below looking at examples of it.
Multiplication has several special cases that are worth considering as
operations in themselves. We’ve already seen one of these: When F(z) = z”’
we get the shifting operation (7.14). In that case the sum h,, becomes the
single term gnPm, because all fk's ue 0 except for fm = 1.