Concrete mathematics : a foundation for computer science

5.4 GENERATING FUNCTIONS 199
interesting. But when n is even, say n = 2, we get a nontrivial sum that’s
different from Vandermonde’s convolution:
(ii)(;)-(;)(;)+(;)(;) =2(i)-r’= -?.
So (5.55) checks out fine when n = 2. It turns out that (5.30) is a special case
of our new identity (5.55).
Binomial coefficients also show up in some other generating functions,
most notably the following important identities in which the lower index
stays fixed and the upper index varies:
lfyou have a highlighter
pen, these
1
(1 -Z)n+'
= t(nn+k)zk,
k>O
integern30
two equations have
got to be marked. Zk , integer n 3 0.
(5.56)
(5.57)
The second identity here is just the first one multiplied by zn, that is, “shifted
right” by n places. The first identity is just a special case of the binomial
theorem in slight disguise: If we expand (1 - z)-~-’ by (5.13), the coefficient
of zk is (-“,-‘)(-l)“, which can be rewritten as (kl”) or (n:k) by negating
the upper index. These special cases are worth noting explicitly, because they
arise so frequently in applications.
When n = 0 we get a special case of a special case, the geometric series:
1
- zz
1-z
1 +z+z2 +z3 + . . . = X2".
This is the generating function for the sequence (1 , 1 , 1, . . . ), and it is especially
useful because the convolution of any other sequence with this one is
the sequence of sums: When bk = 1 for all k, (5.54) reduces to
cn = g ak.
k=O
Therefore if A(z) is the generating function for the summands (ao, al , a2, . ),
then A(z)/(l -2) is the generating function for the sums (CO,CI ,cz,. . .).
The problem of derangements, which we solved by inversion in connection
with hats and football fans, can be resolved with generating functions in an
interesting way. The basic recurrence
n! = x 0 L (n-k)i
k
k>O