Concrete mathematics : a foundation for computer science

200 BINOMIAL COEFFICIENTS
can be put into the form of a convolution if we expand (L) in factorials and
divide both sides by n!:
n 1 (n-k)i
1=x-p.
k=O k! (n-k)!
The generating function for the sequence (A, A, A, . . . ) is e’; hence if we let
D(z) = t 3zk,
k>O k!
the convolution/recurrence tells us that
1
~ = e’D(z).
1-z
Solving for D(z) gives
D(z) = &eP = &
Equating coefficients of 2” now tells us that
this is the formula we derived earlier by inversion.
So far our explorations with generating functions have given us slick
proofs of things that we already knew how to derive by more cumbersome
methods. But we haven’t used generating functions to obtain any new results,
except for (5.55). Now we’re ready for something new and more surprising.
There are two families of power series that generate an especially rich
class of binomial coefficient identities: Let us define the generalized binomial
series IBt (z) and the generalized exponential series Et(z) as follows:
T&(z) = t(tk)*-‘;; E,(z) = t(tk+ l)k-’ $. (5.58)
k>O
It can be shown that these functions satisfy the identities
B,(z)‘- -T&(z)-’ = 2;; &t(z)-tln&t(z) = z. (5.59)
In the special case t = 0, we have
k>O
730(z) = 1 fz; &O(Z) = e’;
.