6.042/18.062J
Mathematics
for
Computer
Science
April
7,
2005
Srini
Devadas
and
Eric
Lehman
Lecture
Notes
Generating
Functions
Generating
functions
are
one
of
the
most
surprising,
useful,
and
clever
inventions
in
discrete
math.
Roughly
speaking,
generating
functions
transform
problems
about
se
quences
into
problems
about
functions
.
This
is
great
because
we’ve
got
piles
of
mathe
matical
machinery
for
manipulating
functions.
Thanks
to
generating
functions,
we
can
apply
all
that
machinery
to
problems
about
sequences.
In
this
way,
we
can
use
generating
functions
to
solve
all
sorts
of
counting
problems.
There
is
a
huge
chunk
of
mathematics
concerning
generating
functions,
so
we
will
only
get
a
taste
of
the
subject.
In
this
lecture,
we’ll
put
sequences
in
angle
brackets
to
more
clearly
distinguish
them
from
the
many
other
mathemtical
expressions
ﬂoating
around.
1
Generating
Functions
The
ordinary
generating
function
for
the
infinite
sequence
�
g
0
, g
1
, g
2
, g
3
. . .
�
is
the
formal
power
series:
3
G
(
x
) =
g
0
+
g
1
x
+
g
2
x
2
+
g
3
x
+
· · ·
A
generating
function
is
a
“formal”
power
series
in
the
sense
that
we
usually
regard
x
as
a
placeholder
rather
than
a
number.
Only
in
rare
cases
will
we
let
x
be
a
real
number
and
actually
evaluate
a
generating
function,
so
we
can
largely
forget
about
questions
of
convergence.
Not
all
generating
functions
are
ordinary,
but
those
are
the
only
kind
we’ll
consider
here.
Throughout
the
lecture,
we’ll
indicate
the
correspondence
between
a
sequence
and
its
generating
function
with
a
doublesided
arrow
as
follows:
2
3
+
g
3
x
�
g
0
, g
1
, g
2
, g
3
, . . .
�
←→
g
0
+
g
1
x
+
g
2
x
+
· · ·
For
example,
here
are
some
sequences
and
their
generating
functions:
3
�
0
,
0
,
0
,
0
, . . .
�
←→
0 + 0
x
+ 0
x
2
+ 0
x
=
0
+
· · ·
3
�
1
,
0
,
0
,
0
, . . .
�
←→
1 + 0
x
+ 0
x
2
+ 0
x
=
1
+
· · ·
3
�
3
,
2
,
1
,
0
, . . .
�
←→
3 + 2
x
+ 1
x
2
+ 0
x
=
3 + 2
x
+
x
2
+
· · ·
The
pattern
here
is
simple:
the
i
th
term
in
the
sequence
(indexing
from
0)
is
the
coefficient
of
x
i
in
the
generating
function.
Recall
that
the
sum
of
an
infinite
geometric
series
is:
1
3
1 +
z
+
z
2
+
z
=
+
· · ·
1
−
z
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2
Generating
Functions
This
equation
does
not
hold
when
z
≥
1
,
but
once
again
we
won’t
worry
about
conver
 
gence
issues.
This
formula
gives
closedform
generating
functions
for
a
whole
range
of
sequences.
For
example:
1
,
1
,
1
,
1
, . . .
�
1 +
x
+
x
2
+
x
=
�
1
←→
3
+
· · ·
1
−
x
1
4
=
�
1
,
−
1
,
1
,
−
1
, . . .
�
−
x
3
+
x
←→
1
−
x
+
x
2
− · · ·
1 +
x
1
3
3
�
1
, a, a
2
, a , . . .
�
1 +
ax
+
a
2
x
2
+
a x
=
1
−
ax
←→
3
+
· · ·
1
4
+
x
6
�
1
,
0
,
1
,
0
,
1
,
0
, . . .
�
1 +
x
2
+
x
=
1
−
x
2
←→
+
· · ·
2
Operations
on
Generating
Functions
The
magic
of
generating
functions
is
that
we
can
carry
out
all
sorts
of
manipulations
on
sequences
by
performing
mathematical
operations
on
their
associated
generating
func
tions.
Let’s
experiment
with
various
operations
and
characterize
their
effects
in
terms
of
sequences.
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 Spring '11
 Dr.EricLehman
 Computer Science, Recurrence relation, Fibonacci number, F3

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