Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 694 — #702
694
Chapter 16
Generating Functions
In the ring of formal power series, equation (16.20) implies that the zero sequence
Z has no inverse, so 1=Z is undefined—just as the expression 1/0 is undefined
over the real numbers or the ring Z n of Section 9.7.1. It’s not hard to verify
that a series has an inverse iff its initial element is nonzero (see Problem 16.25).
Now we can explain the proper way to understand a generating function definition
1X
G.x/ WWD g n x n :
nD0
It simply means that G.x/ really refers to its infinite sequence of coefficients .g 0 ; g 1 ; : : : /
in the ring of formal power series. The simple expression x can be understood as
referring to the sequence
X WWD .0; 1; 0; 0; : : : ; 0; : : : /:
Likewise, 1
x abbreviates the sequence J above, and the familiar equation
1
1 x D 1 C x C x2 C x 3 C (16.21)
can be understood as a way of restating the assertion that K is 1=J . In other words,
the powers of the variable x just serve as a place holders—and as reminders of the
definition of convolution. The equation (16.21) has nothing to do with the values
of x or the convergence of the series. Rather, it is stating a property that holds in
the ring of formal power series. The reasoning about the divergent series in the
previous section is completely justified as properties of formal power series.
16.6 References
[47], [23], [9] [18]
Problems for Section 16.1
Practice Problems
Problem 16.1.
The notation Œx n F .x/ refers to the coefficient of x n in the generating function