
Basic Tools 33
Example 2.32 The generating function for the sequence {n − m}
n,m≥0
is given by
n,m≥0
(n − m)x
n
y
m
=
n,m≥0
nx
n
y
m
−
n,m≥0
mx
n
y
m
= x
∂
∂x
n,m≥0
x
n
y
m
− y
∂
∂y
n,m≥0
x
n
y
m
= x
∂
∂x
1
(1 −x)(1 −y)
− y
∂
∂y
1
(1 − x)(1 − y)
=
x − y
(1 − x)
2
(1 − y)
2
.
Now, let us discuss how one can obtain the generating function of a new sequence from
the generating function of a related sequence. For instance, what is the generating function
for the sequence {a
n+1
}
n≥0
, when the generating function for the sequence {a
n
}
n≥0
is
known. There exist several well-known rules and we present a selection of them, following
[1145, Chapter 2]. We denote the derivative also by D =
d
dx
.
Rule 2.33 Le