formal power series

Laurent series

Formal power series allow one to employ much of the analyticalmachinery of power series in settings which don’t have natural notionsof convergence. They are also useful in order to compactly describesequences and to find closed formulasfor recursively described sequences; this is known as the method ofgenerating functions and will be illustrated below.

We start with a commutative ring $R$. We want to define the ring offormal 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 aninfinite sum of the form ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{X}^n$, where thecoefficients $a_n$ are elements of $R$; any choice of coefficients$a_n$ is allowed. $R[[X]]$ is actually a topological ring so thatthese infinite sums are well-defined and convergent. The addition andmultiplication of such sums follows the usual laws of power series.

Formal construction

Start with the set $R^{\mathbb{N}}$ of all infinite sequences in $R$.Define addition of two such sequences by

$$(a_n)+(b_n)=(a_n+b_n)$$

and multiplication by

$$(a_n)(b_n)=(\sum _{k=0}^n{a}_k{b}_{n-k}).$$

This turns $R^{\mathbb{N}}$ into a commutative ring with multiplicativeidentity (1,0,0,…). We identify the element $a$ of $R$ with thesequence ($a$,0,0,…) and define $X:=(0,1,0,0,\mathrm{\dots })$. Then every elementof $R^{\mathbb{N}}$ of the form $(a_0,a_1,a_2,\mathrm{\dots },a_N,0,0,\mathrm{\dots })$ can be written as the finite sum

$$\sum _{n=0}^N{a}_n{X}^n.$$

In order to extend this equation to infinite series, we need a metricon $R^{\mathbb{N}}$. We define $d((a_n),(b_n))=2^{-k}$, where $k$ is thesmallest natural number such that $a_k\ne b_k$ (if there is not such$k$, then the two sequences are equal and we define their distance tobe zero). This is a metric which turns $R^{\mathbb{N}}$ into a topologicalring, and the equation

$$(a_n)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n$$

can now be rigorously proven using the notion of convergence arisingfrom $d$; in fact, any rearrangement of the series converges to thesame limit.

This topological ring is the ring of formal power series over $R$ andis 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 thesequences which end in zeros.

The geometric series formula is valid in $R[[X]]$:

$${(1-X)}^{-1}=\sum _{n=0}^{\mathrm{\infty }}{X}^n$$

An element $\sum a_n{X}^n$ of $R[[X]]$ is invertible in $R[[X]]$ if and only if itsconstant coefficient $a_0$ is invertible in $R$ (see invertible formal power series).  This implies that theJacobson radical of $R[[X]]$ is the ideal generated by $X$ and theJacobson 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 $R[[X]]$ isequal to the product topology on $R^{\mathbb{N}}$ where $R$ is equippedwith 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 $I=(X)$ isthe ideal generated by $X$ (whose elements are precisely the formalpower series with zero constant coefficient).

If $R=K$ is a field, we can consider the quotient field of theintegral domain $K[[X]]$; it is denoted by $K((X))$ and called a (formal) power series field. It is atopological field whose elements are called formal Laurentseries; they can be uniquely written in the form

$$f=\sum _{n=-M}^{\mathrm{\infty }}{a}_n{X}^n$$

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 withvalues in the real or complex numbers. Formal power series can also beinterpreted as functions, but one has to be careful with the domainand codomain. If $f=\sum a_n{X}^n$ is an element of $R[[X]]$, if $S$ is acommutative associative algebra over $R$, if $I$ an ideal in $S$ suchthat the $I$-adic topology on $S$ is complete, and if $x$ is an elementof $I$, then we can define

$$f(x):=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n.$$

This latter series is guaranteed to converge in $S$ given the aboveassumptions. Furthermore, we have

$$(f+g)(x)=f(x)+g(x)$$

and

$$(fg)(x)=f(x)g(x)$$

(unlike in the case of bona fide functions, these formulas are notdefinitions 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 powerseries, 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 anyformal power series $f\in R[[X]]$.

With this formalism, we can give an explicit formula for themultiplicative inverse of a power series $f$ whose constantcoefficient $a=f(0)$ is invertible in $R$:

$$f^{-1}=\sum _{n=0}^{\mathrm{\infty }}{a}^{-n-1}{(a-f)}^n$$

Differentiating formal power series

If $f={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{X}^n\in R[[X]]$, we define the formal derivative of $f$ as

$$\mathrm{D}f=\sum _{n=1}^{\mathrm{\infty }}{a}_{n}n{X}^{n-1}.$$

This operation is $R$-linear, obeys the product rule

$$\mathrm{D}(f\cdot g)=(\mathrm{D}f)\cdot g+f\cdot (\mathrm{D}g)$$

and the chain rule:

$$\mathrm{D}(f(g))=(\mathrm{D}f)(g)\cdot \mathrm{D}g$$

(in case g(0)=0).

In a sense, all formal power series are Taylor series, because if$f=\sum a_n{X}^n$, then

$$({\mathrm{D}}^{k}f)(0)=k!a_k$$

(here $k!$ denotes the element $1\times (1+1)\times (1+1+1)\times \mathrm{\dots }\in R$.

One can also define differentiation for formal Laurent series in anatural way, and then the quotient rule, in addition to the ruleslisted above, will also be valid.

Power series in several variables

The fastest way to define the ring $R[[X_1,\mathrm{\dots },X_r]]$ of formal powerseries over $R$ in $r$ variables starts with the ring $S=R[X_1,\mathrm{\dots },X_r]$ of polynomials over $R$. Let $I$ be the ideal in $S$generated by $X_1,\mathrm{\dots },X_r$, consider the $I$-adic topology on $S$, andform its completion. This results in a complete topological ringcontaining $S$ which is denoted by $R[[X_1,\mathrm{\dots },X_r]]$.

For $𝐧=(n_1,\mathrm{\dots },n_r)\in {\mathbb{N}}^r$, we write${𝐗}^{𝐧}=X_1^{n_1}\mathrm{\cdots }{X}_r^{n_r}$. Then every element of$R[[X_1,\mathrm{\dots },X_r]]$ can be written in a unique was as a sum

$$\sum _{𝐧\in {\mathbb{N}}^r}{a}_{𝐧}{𝐗}^{𝐧}$$

where the sum extends over all $𝐧\in {\mathbb{N}}^r$. These sums convergefor any choice of the coefficients $a_{𝐧}\in R$ and the orderin which the summation is carried out does not matter.

If $J$ is the ideal in $R[[X_1,\mathrm{\dots },X_r]]$ generated by $X_1,\mathrm{\dots },X_r$(i.e. $J$ consists of those power series with zero constantcoefficient), then the topology on $R[[X_1,\mathrm{\dots },X_r]]$ is the $J$-adictopology.

Since $R[[X_1]]$ is a commutative ring, we can define its powerseries ring, say $R[[X_1]][[X_2]]$. This ring is naturally isomorphic tothe ring $R[[X_1,X_2]]$ just defined, but as topological rings the twoare different.

If $K=R$ is a field, then $K[[X_1,\mathrm{\dots },X_r]]$ is a unique factorizationdomain.

Similar to the situation described above, we can “apply” powerseries in several variables to other power series with zero constantcoefficients. It is also possible to define partial derivatives forformal power series in a straightforward way. Partial derivativescommute, as they do for continuously differentiable functions.

Uses

One can use formal power series to prove several relations familarfrom analysis in a purely algebraic setting. Consider for instance thefollowing elements of $\mathbb{Q}[[X]]$:

$$\sin (X):=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{(2n+1)!}{X}^{2n+1}$$

$$\cos (X):=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{(2n)!}{X}^{2n}$$

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 $\mathbb{Q}[[X,Y]]$).

As an example of the method of generating functions, consider theproblem 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 thering $ℝ[[X]]$ and define the power series

$$f=\sum _{n=0}^{\mathrm{\infty }}{f}_n{X}^n;$$

$f$ is called the generating function for the sequence $(f_n)$.The generating function for the sequence $(f_{n-1})$ is $Xf$ whilethat for $(f_{n-2})$ is $X^{2}f$. From the recurrence relation, wetherefore see that the power series $Xf+X^{2}f$ agrees with $f$ exceptfor the first two coefficients. Taking these into account, we findthat

$$f=Xf+X^{2}f+X$$

(this is the crucial step; recurrence relations can almost always betranslated into equations for the generating functions). Solving thisequation for $f$, we get

$$f=\frac{X}{1-X-X^2}.$$

Using the golden ratio ${\varphi }_1=(1+\sqrt{5})/2$ and${\varphi }_2=(1-\sqrt{5})/2$, we can write the latter expression as

$$\frac{1}{\sqrt{5}}\left(\frac{1}{1-{\varphi }_{1}X}-\frac{1}{1-{\varphi }_{2}X}\right).$$

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

$$f_n=\frac{1}{\sqrt{5}}\left({\varphi }_1^n-{\varphi }_2^n\right).$$

In algebra, the ring $K[[X_1,\mathrm{\dots },X_r]]$ (where $K$ is a field) is oftenused as the “standard, most general” complete local ring over $K$.

Universal property

The power series ring $R[[X_1,\mathrm{\dots },X_r]]$ can be characterized by thefollowing universal property: if $S$ is a commutative associativealgebra over $R$, if $I$ is an ideal in $S$ such that the $I$-adictopology on $S$ is complete, and if $x_1,\mathrm{\dots },x_r\in I$ are given, thenthere exists a unique $\mathrm{\Phi }:R[[X_1,\mathrm{\dots },X_r]]\to S$ with thefollowing properties:

• $\mathrm{\Phi }$ is an $R$-algebra homomorphism

• $\mathrm{\Phi }$ is continuous (http://planetmath.org/Continuous)

• $\mathrm{\Phi }(X_i)=x_i$ for $i=1,\mathrm{\dots },r$.

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