Formal power series

Formal power series are devices in mathematics which allow 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 find closed formulas for recursively defined sequences; this is known as the method of generating functions.

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 Xⁿ where the coefficients a_n are elements of R. R[[X]] is actually a topological ring so that this infinite sum is well-defined. The addition and multiplications of such sums follow 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) = (∑_0≤k≤n a_k b_n-k)

(compare convolution). 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 (a₀, a₁, a₂,...,a_N,0,0,...) can be written as the finite sum

∑_0≤n≤N a_n Xⁿ.

In order to extend this expansion 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 no such k, then the two sequences are the same 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 Xⁿ

can now be rigorously proven using the notion of convergence arising from d; in fact, any rearrangement of the series converges to the same value.

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 Xⁿ

An element ∑ a_n Xⁿ of R[[X]] is invertible if and only if a₀ is invertible in R. This implies that the Jacobson radical of R[[X]] is the ideal generated by X and the Jacobson radical of R.

The maximal ideals of R[[X]] all arise from those in R in the following manner: an ideal M of R[[X]] is maximal if and only if M ∩ R is a maximal ideal of R and M is generated as an ideal by X and M ∩ 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 when R is equipped with the discrete topology. It follows that R[[X]] is compact if and ony 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 (which consists of all formal power series whose zeroth coefficient is zero).

Formal power series as functions

In analysis, every convergenet 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 Xⁿ is an element of R[[X]], S is a commutative associative algebra over R, I is an ideal in S such that the I-adic topology on S is complete, and u is an element of I, then we can define

f(u) = ∑φ(a_n) uⁿ.

This latter series is guaranteed to converge in S given the above assumptions on u. Furthermore, we have

(f+g)(u) = f(u) + g(u)

and

(fg)(u) = f(u) g(u)

(unlike in the case of functions, these formulas are not definitions but have to be 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.

Power series in serveral variables

To define the ring R[[X₁,...,X_r]] of formal power series in r variables, we start with the set R^{N^r} of all functions from N^r to R. Two such functions f and g are added by

(f + g)(n) = f(n) + g(n) for all n in N^r

and multiplied by

(fg)(n) = ∑_m+k=n f(m)g(k) for all n in N^r

where the sum extends over the (finitely many) pairs of elements of N^r which add up to n. We then have constructed a commutative ring; let's call it S for now.

Define elements e₁=(1,0,...,0), e₂=(0,1,...,0), ..., e_r=(0,0,...,1) in N^r and define elements X₁,...,X_r in S by

X_i(e_i) = 1 and X_i(n) = 0 for n ≠ e_i.

Let I be the ideal in S generated by X₁,...,X_r and equip S with the I-adic topology. For n=(n₁,...,n_r)∈N^r, we write Xⁿ = X₁^n₁...X_r^n_r. Then every element of S can be written in a unique way as a sum

∑_n a_n Xⁿ

where the sum extends over the (infinitely many) elements of N^r and a_{n_{∈R. The order in which the elements are added doesn't matter.}}

The standard notation for S is R[[X₁,...,X_r]]; it is an associative algebra over R, complete and Hausdorff as a topological ring.

Since R[[X₁]] is a commutative ring, we can define its power series ring, say R[[X₁]][[X₂]]. This ring is isomorphic to the ring R[[X₁,X₂]] just defined, but as topological rings the two are different.

If R is a field, then R[[X₁,...,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.

Universal property

The power series ring R[[X₁,...,X_r]] can be characterized by the following universal property: if S is a commutative ring with ideal I such that the I-adic topology on S is complete, if φ : R -> S is a ring homomorphism and if x₁,...,x_r are elements of I, then there is a unique Φ : R[[X₁,...,X_n]] -> S with the following properties:

Φ is a ring homomorphism
Φ is continuous
Φ(a) = φ(a) for each a∈R
Φ(X_i) = x_i for i=1...r.