formal power series

formal power series

\PMlinkescapephrase

Laurent series

Formal power series

allow one to employ much of the analytical
machinery of

power series

in settings which don’t have natural notions
of convergence. They are also useful in order to compactly describe

sequences

and to find

closed formulas

for recursively described sequences; this is known as the method of
generating functions and will be illustrated below.

We start with a

commutative ring

R

. We want to define the ring of
formal power series over

R

in the variable

X

, denoted by

R

⁢

[

[

X

]

]

;
each element of this ring can be written in a unique way as an
infinite sum of the form

∑

n

=

0

∞

a

n

⁢

X

n

, where the

coefficients

a

n

are elements of

R

; any choice of coefficients

a

n

is allowed.

R

⁢

[

[

X

]

]

is actually a

topological ring

so that
these infinite sums are well-defined and

convergent

. The

addition

and

multiplication

of such sums follows the usual laws of power series.

Formal construction

Start with the set

R

ℕ

of all

infinite

sequences in

R

.
Define addition of two such sequences by

(

a

n

)

+

(

b

n

)

=

(

a

n

+

b

n

)

and multiplication by

(

a

n

)

⁢

(

b

n

)

=

(

∑

k

=

0

n

a

k

⁢

b

n

-

k

)

.

This turns

R

ℕ

into a commutative ring with

multiplicative
identity

(1,0,0,…). We identify the

element

a

of

R

with the
sequence (

a

,0,0,…) and define

X

:=

(

0

,

1

,

0

,

0

,

…

)

. Then every element
of

R

ℕ

of the form

(

a

0

,

a

1

,

a

2

,

…

,

a

N

,

0

,

0

,

…

)

can be written as the

finite

sum

∑

n

=

0

N

a

n

⁢

X

n

.

In order to extend this

equation

to infinite series, we need a metric
on

R

ℕ

. We define

d

⁢

(

(

a

n

)

,

(

b

n

)

)

=

2

-

k

, where

k

is the
smallest

natural number

such that

a

k

≠

b

k

(if there is not such

k

, then the two sequences are equal and we define their

distance

to
be zero). This is a metric which turns

R

ℕ

into a topological
ring, and the equation

(

a

n

)

=

∑

n

=

0

∞

a

n

⁢

X

n

can now be rigorously proven using the notion of convergence arising
from

d

; in fact, any rearrangement of the series

converges

to the
same limit.

This topological ring is the ring of formal power series over

R

and
is denoted by

R

⁢

[

[

X

]

]

.

Properties

R

⁢

[

[

X

]

]

is an

associative algebra

over

R

which contains the ring

R

⁢

[

X

]

of

polynomials

over

R

; the polynomials correspond to the
sequences which end in zeros.

The geometric series

formula

is valid in

R

⁢

[

[

X

]

]

:

(

1

-

X

)

-

1

=

∑

n

=

0

∞

X

n

An element

∑

a

n

⁢

X

n

of

R

⁢

[

[

X

]

]

is

invertible

in

R

⁢

[

[

X

]

]

if and only if its
constant coefficient

a

0

is invertible in

R

(see

invertible formal power series

).  This implies that the

Jacobson radical

of

R

⁢

[

[

X

]

]

is the

ideal generated by

X

and the
Jacobson radical of

R

.

Several

algebraic

properties of

R

are inherited by

R

⁢

[

[

X

]

]

:

•

if

R

is a

local ring

, then so is

R

⁢

[

[

X

]

]

•

if

R

is

Noetherian

, then so is

R

⁢

[

[

X

]

]

•

if

R

is an

integral domain

, then so is

R

⁢

[

[

X

]

]

•

if

R

is a field, then

R

⁢

[

[

X

]

]

is a

discrete valuation ring

.

The metric space

(

R

⁢

[

[

X

]

]

,

d

)

is

complete

. The

topology

on

R

⁢

[

[

X

]

]

is
equal to the

product topology

on

R

ℕ

where

R

is equipped
with the

discrete topology

. It follows from

Tychonoff

’s

theorem

that

R

⁢

[

[

X

]

]

is

compact

if and only if

R

is finite. The topology on

R

⁢

[

[

X

]

]

can also be seen as the

I

-adic topology, where

I

=

(

X

)

is
the ideal generated by

X

(whose elements are precisely the formal
power series with zero constant coefficient).

If

R

=

K

is a field, we can consider the

quotient field

of the
integral domain

K

⁢

[

[

X

]

]

; it is denoted by

K

⁢

(

(

X

)

)

and called a (

formal

)

power series field

. It is a
topological field whose elements are called

formal Laurent
series

; they can be uniquely written in the form

f

=

∑

n

=

-

M

∞

a

n

⁢

X

n

where

M

is an

integer

which depends on the Laurent series

f

.

Formal power series as functions

In

analysis

, every convergent power series defines a

function

with
values in the real or

complex numbers

. Formal power series can also be
interpreted as functions, but one has to be careful with the domain
and codomain. If

f

=

∑

a

n

⁢

X

n

is an element of

R

⁢

[

[

X

]

]

, if

S

is a

commutative

associative algebra over

R

, if

I

an ideal in

S

such
that the

I

-adic topology on

S

is complete, and if

x

is an element
of

I

, then we can define

f

⁢

(

x

)

:=

∑

n

=

0

∞

a

n

⁢

x

n

.

This latter series is guaranteed to converge in

S

given the above

assumptions

. Furthermore, we have

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

and

(

f

⁢

g

)

⁢

(

x

)

=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

(unlike in the case of bona fide functions, these formulas are not
definitions but have to proved).

Since the topology on

R

⁢

[

[

X

]

]

is the

(

X

)

-adic topology and

R

⁢

[

[

X

]

]

is complete, we can in particular apply power series to other power
series, provided that the

arguments

don’t have constant coefficients:

f

⁢

(

0

)

,

f

⁢

(

X

2

-

X

)

and

f

⁢

(

(

1

-

X

)

-

1

-

1

)

are all well-defined for any
formal power series

f

∈

R

⁢

[

[

X

]

]

.

With this

formalism

, we can give an

explicit formula

for the

multiplicative inverse

of a power series

f

whose constant
coefficient

a

=

f

⁢

(

0

)

is invertible in

R

:

f

-

1

=

∑

n

=

0

∞

a

-

n

-

1

⁢

(

a

-

f

)

n

Differentiating formal power series

If

f

=

∑

n

=

0

∞

a

n

⁢

X

n

∈

R

⁢

[

[

X

]

]

, we define the formal derivative of

f

as

D

⁡

f

=

∑

n

=

1

∞

a

n

⁢

n

⁢

X

n

-

1

.

This

operation

is

R

-linear, obeys the

product rule

D

⁡

(

f

⋅

g

)

=

(

D

⁡

f

)

⋅

g

+

f

⋅

(

D

⁡

g

)

and the

chain rule

:

D

⁡

(

f

⁢

(

g

)

)

=

(

D

⁡

f

)

⁢

(

g

)

⋅

D

⁡

g

(in case g(0)=0).

In a sense, all formal power series are

Taylor series

, because if

f

=

∑

a

n

⁢

X

n

, then

(

D

k

⁡

f

)

⁢

(

0

)

=

k

!

⁢

a

k

(here

k

!

denotes the element

1

×

(

1

+

1

)

×

(

1

+

1

+

1

)

×

…

∈

R

.

One can also define

differentiation

for formal Laurent series in a
natural way, and then the

quotient rule

, in addition to the rules
listed above, will also be valid.

Power series in several variables

The fastest way to define the ring

R

⁢

[

[

X

1

,

…

,

X

r

]

]

of formal power
series over

R

in

r

variables starts with the ring

S

=

R

⁢

[

X

1

,

…

,

X

r

]

of polynomials over

R

. Let

I

be the ideal in

S

generated by

X

1

,

…

,

X

r

, consider the

I

-adic topology on

S

, and
form its

completion

. This results in a complete topological ring
containing

S

which is denoted by

R

⁢

[

[

X

1

,

…

,

X

r

]

]

.

For

𝐧

=

(

n

1

,

…

,

n

r

)

∈

ℕ

r

, we write

𝐗

𝐧

=

X

1

n

1

⁢

⋯

⁢

X

r

n

r

. Then every element of

R

⁢

[

[

X

1

,

…

,

X

r

]

]

can be written in a unique was as a sum

∑

𝐧

∈

ℕ

r

a

𝐧

⁢

𝐗

𝐧

where the sum extends over all

𝐧

∈

ℕ

r

. These sums converge
for any choice of the coefficients

a

𝐧

∈

R

and the order
in which the

summation

is carried out does not matter.

If

J

is the ideal in

R

⁢

[

[

X

1

,

…

,

X

r

]

]

generated by

X

1

,

…

,

X

r

(i.e.

J

consists of those power series with zero constant
coefficient), then the topology on

R

⁢

[

[

X

1

,

…

,

X

r

]

]

is the

J

-adic
topology.

Since

R

⁢

[

[

X

1

]

]

is a commutative ring, we can define its power
series ring, say

R

⁢

[

[

X

1

]

]

⁢

[

[

X

2

]

]

. This ring is naturally

isomorphic

to
the ring

R

⁢

[

[

X

1

,

X

2

]

]

just defined, but as topological rings the two
are different.

If

K

=

R

is a field, then

K

⁢

[

[

X

1

,

…

,

X

r

]

]

is a

unique factorization
domain

.

Similar

to the situation described above, we can “apply” power
series in several variables to other power series with zero constant
coefficients. It is also possible to define

partial derivatives

for
formal power series in a straightforward way. Partial derivatives
commute, as they do for

continuously differentiable functions

.

Uses

One can use formal power series to prove several

relations

familar
from analysis in a purely algebraic setting. Consider for

instance

the
following elements of

ℚ

⁢

[

[

X

]

]

:

sin

⁡

(

X

)

:=

∑

n

=

0

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

X

2

⁢

n

+

1

cos

⁡

(

X

)

:=

∑

n

=

0

∞

(

-

1

)

n

(

2

⁢

n

)

!

⁢

X

2

⁢

n

Then one can easily show that

sin

2

⁡

(

X

)

+

cos

2

⁡

(

X

)

=

1

and

D

⁢

sin

=

cos

as well as

sin

⁡

(

X

+

Y

)

=

sin

⁡

(

X

)

⁢

cos

⁡

(

Y

)

+

cos

⁡

(

X

)

⁢

sin

⁡

(

Y

)

(the latter being valid in the ring

ℚ

⁢

[

[

X

,

Y

]

]

).

As an example of the method of generating functions, consider the
problem of finding a closed formula for the

Fibonacci numbers

f

n

defined by

f

n

+

2

=

f

n

+

1

+

f

n

,

f

0

=

0

, and

f

1

=

1

. We work in the
ring

ℝ

⁢

[

[

X

]

]

and define the power series

f

=

∑

n

=

0

∞

f

n

⁢

X

n

;

f

is called the

generating function

for the sequence

(

f

n

)

.
The generating function for the sequence

(

f

n

-

1

)

is

X

⁢

f

while
that for

(

f

n

-

2

)

is

X

2

⁢

f

. From the

recurrence relation

, we
therefore see that the power series

X

⁢

f

+

X

2

⁢

f

agrees with

f

except
for the first two coefficients. Taking these into account, we find
that

f

=

X

⁢

f

+

X

2

⁢

f

+

X

(this is the crucial step; recurrence relations can almost always be
translated into equations for the generating functions). Solving this
equation for

f

, we get

f

=

X

1

-

X

-

X

2

.

Using the

golden ratio

ϕ

1

=

(

1

+

5

)

/

2

and

ϕ

2

=

(

1

-

5

)

/

2

, we can write the latter expression as

1

5

⁢

(

1

1

-

ϕ

1

⁢

X

-

1

1

-

ϕ

2

⁢

X

)

.

These two power series are known explicitly because they are geometric
series; comparing coefficients, we find the explicit formula

f

n

=

1

5

⁢

(

ϕ

1

n

-

ϕ

2

n

)

.

In

algebra

, the ring

K

⁢

[

[

X

1

,

…

,

X

r

]

]

(where

K

is a field) is often
used as the “standard, most general” complete local ring over

K

.

Universal property

The power series ring

R

⁢

[

[

X

1

,

…

,

X

r

]

]

can be characterized by the
following

universal property

: if

S

is a commutative associative
algebra over

R

, if

I

is an ideal in

S

such that the

I

-adic
topology on

S

is complete, and if

x

1

,

…

,

x

r

∈

I

are given, then
there exists a

unique

Φ

:

R

⁢

[

[

X

1

,

…

,

X

r

]

]

→

S

with the
following properties:

•

Φ

is an

R

-algebra homomorphism

•

Φ

is

continuous

(

http://planetmath.org/Continuous

)

•

Φ

⁢

(

X

i

)

=

x

i

for

i

=

1

,

…

,

r

.

Title

formal power series

Canonical name

FormalPowerSeries

Date of creation

2013-03-22 12:49:30

Last modified on

2013-03-22 12:49:30

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

14

Author

AxelBoldt (56)

Entry type

Topic

Classification

msc 13H05

Classification

msc 13B35

Classification

msc 13J05

Classification

msc 13F25

Related topic

PowerSeries

Related topic

SumOfKthPowersOfTheFirstNPositiveIntegers

Related topic

PolynomialRingOverIntegralDomain

Related topic

FiniteRingHasNoProperOverrings

Defines

formal power series

Defines

generating function

Defines

formal Laurent series

Defines

power series field

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