In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values. One way to view a formal power series is as an infinite ordered sequence of numbers. In this case, the powers of the variable are used only to indicate the order of the coefficients, so that the coefficient of
Contents
- Introduction
- The ring of formal power series
- Definition of the formal power series ring
- Ring structure
- Topological structure
- Alternative topologies
- Universal property
- Operations on formal power series
- Power series raised to powers
- Inverting series
- Dividing series
- Extracting coefficients
- Composition of series
- Example
- Composition inverse
- Formal differentiation of series
- Algebraic properties of the formal power series ring
- Topological properties of the formal power series ring
- Applications
- Interpreting formal power series as functions
- Formal Laurent series
- Formal residue
- The Lagrange inversion formula
- Power series in several variables
- Topology
- Operations
- Non commuting variables
- On a semiring
- Replacing the index set by an ordered abelian group
- Examples and related topics
- References
Introduction
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms. Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value). For example, consider the series
If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients [1, −3, 5, −7, 9, −11, ...]. In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the factorials [1, 1, 2, 6, 24, 120, 720, 5040, … ] as coefficients, even though the corresponding power series diverges for any nonzero value of X.
Arithmetic on formal power series is carried out by simply pretending that the series are polynomials. For example, if
then we add A and B term by term:
We can multiply formal power series, again just by treating them as polynomials (see in particular Cauchy product):
Notice that each coefficient in the product AB only depends on a finite number of coefficients of A and B. For example, the X5 term is given by
For this reason, one may multiply formal power series without worrying about the usual questions of absolute, conditional and uniform convergence which arise in dealing with power series in the setting of analysis.
Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series A is a formal power series C such that AC = 1, provided that such a formal power series exists. It turns out that if A has a multiplicative inverse, it is unique, and we denote it by A−1. Now we can define division of formal power series by defining B/A to be the product BA−1, provided that the inverse of A exists. For example, one can use the definition of multiplication above to verify the familiar formula
An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator for a formal power series in one variable extracts the coefficient of
Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.
The ring of formal power series
The set of all formal power series in X with coefficients in a commutative ring R form another ring that is written R[[X]], and called the ring of formal power series in the variable X over R.
Definition of the formal power series ring
One can characterize
Ring structure
As a set,
and multiplication by
This type of product is called the Cauchy product of the two sequences of coefficients, and is a sort of discrete convolution. With these operations,
The product is in fact the same one used to define the product of polynomials in one indeterminate, which suggests using a similar notation. One embeds
these are precisely the polynomials in
and
which is quite convenient, but one must be aware of the distinction between formal summation (a mere convention) and actual addition.
Topological structure
Having stipulated conventionally that
one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in
Informally, two sequences
This topological structure, together with the ring operations described above, form a topological ring. This is called the ring of formal power series over
The topological structure allows much more flexible use of infinite summations. For instance the rule for multiplication can be restated simply as
since only finitely many terms on the right affect any fixed
Alternative topologies
The above topology is the finest topology for which
Consider the ring of formal power series
then the topology of above construction only relates to the indeterminate
converges to the power series suggested, which can be written as
would be considered to be divergent, since every term affects the coefficient of
This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives the same topology as one would get by taking formal power series in all inderteminates at once. In the above example that would mean constructing
The same principle could be used to make other divergent limits converge. For instance in
does not exist, so in particular it does not converge to
Universal property
The ring
Operations on formal power series
One can perform algebraic operations on power series to generate new power series. Besides the ring structure operations defined above, we have the following.
Power series raised to powers
If n is a natural number we have
where
for m ≥ 1. (This formula can only be used if m and a0 are invertible in the ring of scalars.)
In the case of formal power series with complex coefficients, the complex powers are well defined at least for series f with constant term equal to 1. In this case, fα can be defined either by composition with the binomial series (1+x)α, or by composition with the exponential and the logarithmic series, fα := exp(αlog(f)), or as the solution of the differential equation f(fα)′ = αfαf′ with constant term 1, the three definitions being equivalent. The rules of calculus (fα)β = fαβ and fαgα = (fg)α easily follow.
Inverting series
The series
in
An important special case is that the geometric series formula is valid in
If
Dividing series
The computation of a quotient
assuming the denominator is invertible (that is,
Extracting coefficients
The coefficient extraction operator applied to a formal power series
in X is written
and extracts the coefficient of Xm, so that
Composition of series
Given formal power series
one may form the composition
where the coefficients cn are determined by "expanding out" the powers of f(X):
Here the sum is extended over all (k, j) with k in N and
A more explicit description of these coefficients is provided by Faà di Bruno's formula, at least in the case where the coefficient ring is a field of characteristic 0.
A point here is that this operation is only valid when
Example
Assume that the ring
then the expression
makes perfect sense as a formal power series. However, the statement
is not a valid application of the composition operation for formal power series. Rather, it is confusing the notions of convergence in
Composition inverse
Whenever a formal series
Formal differentiation of series
Given a formal power series
in R[[X]], we define its formal derivative, denoted Df or f′, by
The symbol D is called the formal differentiation operator. The motivation behind this definition is that it simply mimics term-by-term differentiation of a polynomial.
This operation is R-linear:
for any a, b in R and any f, g in R[[X]]. Additionally, the formal derivative has many of the properties of the usual derivative of calculus. For example, the product rule is valid:
and the chain rule works as well:
whenever the appropriate compositions of series are defined (see above under composition of series).
Thus, in these respects formal power series behave like Taylor series. Indeed, for the f defined above, we find that
where Dk denotes the kth formal derivative (that is, the result of formally differentiating k times).
Algebraic properties of the formal power series ring
The Jacobson radical of
The maximal ideals of
Several algebraic properties of
Topological properties of the formal power series ring
The metric space (R[[X]], d) is complete.
The ring R[[X]] is compact if and only if R is finite. This follows from Tychonoff's theorem and the characterisation of the topology on R[[X]] as a product topology.
Applications
Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions.
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]]:
Then one can show that
The last one being valid in the ring Q[[X,Y]].
For K a field, the ring K[[X1, ..., Xr]] is often used as the "standard, most general" complete local ring over K in algebra.
Interpreting 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
This latter series is guaranteed to converge in S given the above assumptions on X. Furthermore, we have
and
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 (so that they belong to the ideal (X)): f(0), f(X2−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:
If the formal power series g with g(0) = 0 is given implicitly by the equation
f(g) = Xwhere f is a known power series with f(0) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.
Formal Laurent series
A formal Laurent series over a ring
where
which sum is effectively finite because of the assumed vanishing of coefficients at sufficiently negative indices, and which sum zero for sufficiently negative
For a non-zero formal Laurent series, the minimal integer
If
One may define formal differentiation for formal Laurent series in a natural way (term-by-term). Precisely, the formal derivative of the formal Laurent series
which is again an element of
However, in general this is not the case since the factor n for the lowest order term could be equal to 0 in R.
Formal residue
Assume that
is a
The latter shows that the coefficient of
is
Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any
Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to
The Lagrange inversion formula
As mentioned above, any formal series f ∈ K[[X]] with f0 = 0 and f1 ≠ 0 has a composition inverse g in K[[X]]. The following relation between the coefficients of gn and f−k holds ("Lagrange inversion formula"):
In particular, for n = 1 and all k ≥ 1,
Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting it here. Since
Generalizations. One may observe that the above computation can be repeated plainly in more general settings than K((X)): a generalization of the Lagrange inversion formula is already available working in the C((X))-modules XαC((X)), where α is a complex exponent. As a consequence, if f and g are as above, with
For instance, this way one finds the power series for complex powers of the Lambert function.
Power series in several variables
Formal power series in any number of indeterminates (even infinitely many) can be defined. If I is an index set and XI is the set of indeterminates Xi for i∈I, then a monomial Xα is any finite product of elements of XI (repetitions allowed); a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of monomials Xα to a corresponding coefficient cα, and is denoted
and
Topology
The topology on R[[XI]] is such that a sequence of its elements converges only if for each monomial Xα the corresponding coefficient stabilizes. If I is finite, then this the J-adic topology, where J is the ideal of R[[XI]] generated by all the indeterminates in XI. This does not hold if I is infinite. For example, if I = N, then the sequence (fn)n∈N with
As remarked above, the topology on a repeated formal power series ring like R[[X]][[Y]] is usually chosen in such a way that it becomes isomorphic as a topological ring to R[[X,Y]].
Operations
All of the operations defined for series in one variable may be extended to the several variables case.
In the case of the formal derivative, there are now separate partial derivative operators, which differentiate with respect to each of the indeterminates. They all commute with each other.
Universal property
In the several variables case, the universal property characterizing R[[X1, ..., Xr]] becomes the following. If S is a commutative associative algebra over R, if I is an ideal of 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:
Non-commuting variables
The several variable case can be further generalised by taking non-commuting variables Xi for i ∈ I, where I is an index set and then a monomial Xα is any word in the XI; a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of monomials Xα to a corresponding coefficient cα, and is denoted
and multiplication by
where · denotes concatenation of words. These formal power series over R form the Magnus ring over R.
On a semiring
In theoretical computer science, the following definition of a formal power series is given: let Σ be an alphabet (finite set) and S be a semiring. In this context, a formal power series is any mapping r from the set of strings generated by Σ (denoted as Σ∗) to the semiring S. The values of such a mapping r are (somewhat idiosyncratically) denoted as (r, w) were w ∈ Σ∗. Then the mapping r itself is conventionally written as
Replacing the index set by an ordered abelian group
Suppose
for all such I, with
Various properties of
This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.