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Formal power series

Formal power series are devices in mathematics that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of "convergence". They are also useful to compactly describe sequences and to find closed formulas for recursively defined 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 an Xn where the coefficients an are elements of R; any choice of coefficients an 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 follow the usual laws of power series.

Table of contents

Formal construction

Start with the set RN of all infinite sequences in R. Define addition of two such sequences by

<math>
\left( a_n \right) + \left( b_n \right) = \left( a_n + b_n \right) </math>

and multiplication by

<math>
\left( a_n \right) \times \left( b_n \right) = \left( \sum_{k=0}^n a_k b_{n-k} \right) </math>

(compare convolution). This turns RN 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 RN of the form (a0, a1, a2,...,aN,0,0,...) can be written as the finite sum

<math>
\sum_{n=0}^N a_n X^n </math>

In order to extend this expansion to infinite series, we need a metric on RN. We define d((an), (bn)) = 2-k, where k is the smallest natural number such that akbk (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 RN into a topological ring, and the equation

<math>
\left( a_n \right) = \sum_{n \ge 0} a_n X^n </math>

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]]:

<math>
\left( 1 - X \right)^{-1} = \sum_{n \ge 0} X^n </math>

An element ∑ an Xn of R[[X]] is invertible in R[[X]] if and only if its constant coefficient 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 MR is a maximal ideal of R and M is generated as an ideal by X and MR.

Several algebraic properties of R are inherited by R[[X]]:

The metric space (R[[X]], d) is complete. The topology on R[[X]] is equal to the product topology on RN 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 (which consists of all formal power series whose zeroth coefficient is zero).

If K=R is a field, we can consider the quotient field of the integral domain K[[X]]; it is denoted by K((X)). Its elements are formal Laurent series of the form

<math>
f = \sum_{n \ge -M} a_n X^n </math>

where M is an integer which depends on the Laurent series f. K((X)) is a topological field.

Formal power series as functions

In mathematical 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=∑an 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 x is an element of I, then we can define

<math>
f(x) = \sum_{n\ge 0} a_n x^n </math>

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

<math>
(f+g)(x) = f(x) + g(x) </math>

and

<math>
(fg)(x) = f(x) g(x) </math>

Unlike in the case of bona fide 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: f(0), f(X2-X) and f( (1-X)-1 - 1) are all well defined for any formal power series fR[[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:

<math>
f^{-1} = \sum_{n \ge 0} a^{-n-1} (a-f)^n </math>

If the formal power series g with g(0) = 0 is given implicitly by the equation

<math>
f(g) = X </math>

where f is a known power series with f(0) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion theorem.

Differentiating formal power series

If f = ∑ an Xn is an element of R[[X]], we define its formal derivative using the operator D as

<math>
Df = \sum_{n \ge 1} a_n n X^{n-1} </math>

This operation is R-linear:

<math>
D(af + bg) = a Df + b Dg </math>

for a, b in R and f, g in R[[X]].

The formal derivative has many of the properties of the continuous derivative of calculus. For example, the product rule is valid:

<math>
D(fg) = f(Dg) + (Df) g </math>

and the chain rule works as well:

<math>
D(f(u)) = (Df)(u) Du </math>

In a sense, all formal power series are Taylor series, because if f=∑an Xn, then, writing Dk as the kth formal derivative, we find that

<math>
(D_k f)(0) = k! a_k </math>

(here k! is the element 1×(1+1)×(1+1+1)×... of 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[[X1,...,Xr]] of formal power series over R in r variables starts with the ring S = R[X1,...,Xr] of polynomials over R. Let I be the ideal in S generated by X1,...,Xr, 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[[X1,...,Xr]].

For n=(n1,...,nr)∈Nr, we write Xn = X1n1...Xrnr. Then every element of R[[X1,...,Xr]] can be written in a unique way as a sum

<math>
\sum_{\mathbf{n}\in\Bbb{N}^r} a_\mathbf{n} \mathbf{X^n} </math>

These sums converge for any choice of the coefficients anR and the order in which the elements are added doesn't matter.

If J is the ideal in R[[X1,...,Xr]] generated by X1,...,Xr (i.e. J consists of those power series with zero constant coefficients), then the topology on R[[X1,...,Xr]] is the J-adic topology.

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

If K = R is a field, then K[[X1,...,Xr]] 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 familiar from analysis in a purely algebraic setting. Consider for instance the following elements of Q[[X]]:

<math>
\sin(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n+1)!} X^{2n+1} </math>

<math>
\cos(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n)!} X^{2n} </math>

Then one can show that

<math>
\sin^2 + \cos^2 = 1 </math>

and

<math>
D \sin = \cos </math>

as well as

<math>
\sin (X+Y) = \sin(X) \cos(Y) + \cos(X) \sin(Y) </math>

(the latter being valid in the ring Q[[X,Y]]).

As an example of the method of generating functions which arises frequently in combinatorics, consider the problem of finding a closed formula for the Fibonacci numbers fn defined by f0 = 0, f1 = 1, and fn = fn-1 + fn-2 for n≥2. We work in the ring R[[X]] and define the power series

<math>
f = \sum_{n \ge 0} f_n X^n </math>

f is called the generating function for the sequence (fn). The generating function for the sequence (fn-1) is Xf and that of (fn-2) is X2f. From the recurrence relation, we therefore see that the power series Xf + X2f agrees with f except for the first two coefficients. Taking these into account, we find that

<math>
f = Xf + X^2 f + X </math>

(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

<math>
f = \frac{X} {1 - X - X^2} </math>

The denominator can be factored using the golden ratio φ1 = (1+√5)/2 and φ2 = (1-√5)/2, and the technique of partial fraction decomposition[?] yields

<math>
\frac{1 / \sqrt{5}} {1-\phi_1 X} - \frac{1/\sqrt{5}} {1- \phi_2 X} </math>

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

<math>
f_n = \frac{1} {\sqrt{5}} (\phi_1^n - \phi_2^n) </math>

In algebra, the ring K[[X1,...,Xr]] (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[[X1,...,Xr]] 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 x1,...,xr are elements of I, then there is a unique Φ : R[[X1,...,Xn]] -> S with the following properties:

  • Φ is an R-algebra homomorphism
  • Φ is continuous
  • Φ(Xi) = xi for i=1...r.



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