Summer 2007
STAT333
Summary of generating function
The purpose of this part is twofolds. First, given a sequence of real numbers
{
a
n
}
∞
n
=0
, how
to calculate the generating function
A
(
s
). Secondly, given the generating function
A
(
s
), how
to find the coeﬃcients
{
a
n
}
∞
n
=0
.
♣
Generating function: definition
Given a sequence of real numbers
{
a
0
, a
1
, . . . , a
n
}
=
{
a
n
}
∞
n
=0
, define the power series,
A
(
s
) =
a
0
+
a
1
s
+
a
2
s
2
+
· · ·
=
∞
n
=0
a
n
s
n
for every real number
s
for which the above power series converges. Obviously whether
or not the power series converges at a particular
s
depends on the coeﬃcients
{
a
n
}
∞
n
=0
.
Every power series does exactly one of the following three things:
(a)
A
(
s
) converges only for
s
= 0 (not our interest).
(b)
A
(
s
) converges when

s

< R
and diverges when

s

> R
. In this case,
R
is called
the convergence radius for the power series.
(c)
A
(
s
) converges for all the real
s
(corresponds to the case
R
=
∞
).
In the last two cases, we call
A
(
s
)
the generating function
of the sequence
{
a
n
}
∞
n
=0
.
For the generating function, the close form for
A
(
s
) and the radius are both very
important. When you find the close form, please do not forget the convergence radius.
♣
Examples of the commonly used generating functions
(a)
Geometric:
A
(
s
) =
s
k
1
−
s
=
∑
∞
n
=
k
s
n
with radius
R
= 1. Here
a
n
= 0 for
n
= 0
,
1
, . . . , k
−
1 and
a
n
= 1 for
n
≥
k.
(b)
Alternate Geometric:
A
(
s
) =
(
−
1)
k
s
k
1+
s
=
∑
∞
n
=
k
(
−
1)
n
s
n
with radius
R
= 1. Here
a
n
= 0 for
n
= 0
,
1
, . . . , k
−
1 and
a
n
= (
−
1)
n
for
n
≥
k.
(For the first two formulas, try to understand how you can get them by the formula
for geometric series. In general, it is
First term
1
−
common ratio
.)
(c)
Exponential:
A
(
s
) =
e
s
=
∑
∞
n
=0
s
n
n
!
with radius
R
=
∞
. Here
a
n
= 1
/n
!
,
for
n
= 0
,
1
,
2
. . . .
(Try to under the above formula from
∑
∞
n
=0
s
n
e
−
s
n
!
= 1. That is, the summation of
the pmf of Poisson distribution is equal to 1.)
1
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(d)
Binomial:
A
(
s
) = (1 +
s
)
a
=
∑
∞
n
=0
(
a
n
)
s
n
with radius
R
= 1. Here
a
n
= (
a
n
)
,
for
n
= 0
,
1
,
2
. . . .
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 Power Series, Taylor Series, Probability, Probability theory, pn

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