DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1992, 2008.
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Chapter 14
GENERATING FUNCTIONS
14.1. Introduction
Definition.
By the generating function of a sequence
a
0
,a
1
,a
2
,a
3
,...,
we mean the formal power series
a
0
+
a
1
X
+
a
2
X
2
+
a
3
X
3
+
...
=
∞
X
n
=0
a
n
X
n
.
Example 14.1.1.
The generating function of the sequence
±
k
0
²
,
±
k
1
²
,...,
±
k
k
²
,
0
,
0
,
0
,
0
,...
is given by
±
k
0
²
+
±
k
1
²
X
+
...
+
±
k
k
²
X
k
.
This is equal to (1 +
X
)
k
by the Binomial theorem.
Example 14.1.2.
The generating function of the sequence
1
,...,
1

{z
}
k
,
0
,
0
,
0
,
0
,...
is given by
1 +
X
+
X
2
+
X
3
+
...
+
X
k

1
=
1

X
k
1

X
.
Chapter 14 : Generating Functions
page 1 of 6