Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of formula_1 is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring.
In algebraic geometry and commutative algebra, rings of formal power series are especially tractable topologically complete local rings, allowing calculus-like arguments within a purely algebraic framework. They are analogous in many ways to p-adic numbers. Formal power series can be created from Taylor polynomials using formal moduli. 
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 "X"5 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 formula_8 applied to a formal power series formula_9 in one variable extracts the coefficient of the formula_10th power of the variable, so that formula_11 and formula_12. Other examples include
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.
If one considers the set of all formal power series in "X" with coefficients in a commutative ring "R", the elements of this set collectively constitute another ring which is written formula_14 and called the ring of formal power series in the variable "X" over "R".
Definition of the formal power series ring.
One can characterize formula_15 abstractly as the completion of the polynomial ring formula_16 equipped with a particular metric. This automatically gives formula_15 the structure of a topological ring (and even of a complete metric space). But the general construction of a completion of a metric space is more involved than what is needed here, and would make formal power series seem more complicated than they are. 
It is possible to describe formula_15 more explicitly, and define the ring structure and topological structure separately, as follows.
Ring structure.
As a set, formula_15 can be constructed as the set formula_20 of all infinite sequences of elements of formula_21, indexed by the natural numbers (taken to include 0). Designating a sequence whose term at index formula_10 is formula_23 by formula_24, one defines addition of two such sequences by
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, formula_20 becomes a commutative ring with zero element formula_28 and multiplicative identity formula_29.
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 formula_21 into formula_15 by sending any (constant) formula_32 to the sequence formula_33 and designates the sequence formula_34 by formula_35; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as
these are precisely the polynomials in formula_35. Given this, it is quite natural and convenient to designate a general sequence formula_38 by the formal expression formula_39, even though the latter "is not" an expression formed by the operations of addition and multiplication defined above (from which only finite sums can be constructed). This notational convention allows reformulation of the above definitions as
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 formula_20 is defined and a topology on formula_20 is constructed. There are several equivalent ways to define the desired topology.
Informally, two sequences formula_54 and formula_55 become closer and closer if and only if more and more of their terms agree exactly. Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of formula_35 the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient. This is clearly the case for the right hand side of (), regardless of the values formula_23, since inclusion of the term for formula_58 gives the last (and in fact only) change to the coefficient of formula_59. It is also obvious that the limit of the sequence of partial sums is equal to the left hand side.
This topological structure, together with the ring operations described above, form a topological ring. This is called the ring of formal power series over formula_21 and is denoted by formula_15. The topology has the useful property that an infinite summation converges if and only if the sequence of its terms converges to 0, which just means that any fixed power of formula_35 occurs in only finitely many terms.
The topological structure allows much more flexible usage 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 formula_59. Infinite products are also defined by the topological structure; it can be seen that an infinite product converges if and only if the sequence of its factors converges to 1.
Alternative topologies.
The above topology is the finest topology for which 
always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wishes to use to designate particular formal power series. It can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge. This applies in particular when the base ring formula_21 already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.
Consider the ring of formal power series: formula_67; then the topology of above construction only relates to the indeterminate formula_68, since the topology that was put on formula_69 has been replaced by the discrete topology when defining the topology of the whole ring. So
converges to the power series suggested, which can be written as formula_71; however the summation
would be considered to be divergent, since every term affects the coefficient of formula_68 (which coefficient is itself a power series in formula_35). This asymmetry disappears if the power series ring in formula_68 is given the product topology where each copy of formula_69 is given its topology as a ring of formal power series rather than the discrete topology. As a consequence, for convergence of a sequence of elements of formula_67 it then suffices that the coefficient of each power of formula_68 converges to a formal power series in formula_35, a weaker condition than stabilizing entirely; for instance in the second example given here the coefficient of formula_68converges to formula_81, so the whole summation converges to formula_82.
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 indeterminates at once. In the above example that would mean constructing formula_83 and here a sequence converges if and only if the coefficient of every monomial formula_84 stabilizes. This topology, which is also the formula_85-adic topology, where formula_86 is the ideal generated by formula_35 and formula_68, still enjoys the property that a summation converges if and only if its terms tend to 0.
The same principle could be used to make other divergent limits converge. For instance in formula_89 the limit
does not exist, so in particular it does not converge to 
This is because for formula_92 the coefficient formula_93 of formula_94 does not stabilize as formula_95. It does however converge in the usual topology of formula_96, and in fact to the coefficient formula_97 of formula_98. Therefore, if one would give formula_89 the product topology of formula_100 where the topology of formula_96 is the usual topology rather than the discrete one, then the above limit would converge to formula_98. This more permissive approach is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in analysis, while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be. With this topology it would "not" be the case that a summation converges if and only if its terms tend to 0.
Universal property.
The ring formula_15 may be characterized by the following universal property. If formula_104 is a commutative associative algebra over formula_21, if formula_85 is an ideal of formula_104 such that the formula_85-adic topology on formula_104 is complete, and if formula_110 is an element of formula_85, then there is a "unique" formula_112 with the following properties:
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.
For any natural number "n" we have
where
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, formula_119 can be defined either by composition with the binomial series (1+"x")α, or by composition with the exponential and the logarithmic series, formula_120 or as the solution of the differential equation formula_121 with constant term 1, the three definitions being equivalent. The rules of calculus formula_122 and formula_123 easily follow.
Multiplicative inverse.
The series
is invertible in formula_15 if and only if its constant coefficient formula_49 is invertible in formula_21. This condition is necessary, for the following reason: if we suppose that formula_9 has an inverse formula_129 then the constant term formula_130 of formula_131 is the constant term of the identity series, i.e. it is 1. This condition is also sufficient; we may compute the coefficients of the inverse series formula_132 via the explicit recursive formula
An important special case is that the geometric series formula is valid in formula_15:
If formula_136 is a field, then a series is invertible if and only if the constant term is non-zero, i.e. if and only if the series is not divisible by formula_35. This means that formula_138 is a discrete valuation ring with uniformizing parameter formula_35.
Division.
The computation of a quotient formula_140
assuming the denominator is invertible (that is, formula_49 is invertible in the ring of scalars), can be performed as a product formula_143 and the inverse of formula_144, or directly equating the coefficients in formula_145:
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.
Given formal power series
one may form the "composition"
where the coefficients "c""n" are determined by "expanding out" the powers of "f"("X"):
Here the sum is extended over all ("k", "j") with formula_154 and formula_155 with formula_156
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.
Composition is only valid when formula_157 has "no constant term", so that each formula_158 depends on only a finite number of coefficients of formula_157 and formula_160. In other words, the series for formula_161 converges in the topology of formula_15.
Example.
Assume that the ring formula_21 has characteristic 0 and the nonzero integers are invertible in formula_21. If we denote by formula_98 the formal power series
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 formula_15 and convergence in formula_21; indeed, the ring formula_21 may not even contain any number formula_172 with the appropriate properties.
Composition inverse.
Whenever a formal series 
has "f"0 = 0 and "f"1 being an invertible element of "R", there exists a series 
that is the composition inverse of formula_143, meaning that composing formula_143 with formula_144 gives the series representing the identity function formula_178. The coefficients of formula_144 may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity "X" (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients of "g", as well as the coefficients of the (multiplicative) powers of "g".
Formal differentiation.
Given a formal power series
we define its formal derivative, denoted "Df" or "f" ′, by
The symbol "D" is called the formal differentiation operator. This definition 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 formula_183 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 "D""k" denotes the "k"th formal derivative (that is, the result of formally differentiating "k" times).
Formal antidifferentiation.
If formula_21 is a ring with characteristic zero and the nonzero integers are invertible in formula_21, then given a formal power series
we define its formal antiderivative or formal indefinite integral by
for any constant formula_191.
This operation is "R"-linear:
for any "a", "b" in "R" and any "f", "g" in formula_183 Additionally, the formal antiderivative has many of the properties of the usual antiderivative of calculus. For example, the formal antiderivative is the right inverse of the formal derivative: 
for any formula_195.
Properties.
Algebraic properties of the formal power series ring.
formula_15 is an associative algebra over formula_21 which contains the ring formula_16 of polynomials over formula_21; the polynomials correspond to the sequences which end in zeros.
The Jacobson radical of formula_15 is the ideal generated by formula_35 and the Jacobson radical of formula_21; this is implied by the element invertibility criterion discussed above.
The maximal ideals of formula_15 all arise from those in formula_21 in the following manner: an ideal formula_205 of formula_15 is maximal if and only if formula_207 is a maximal ideal of formula_21 and formula_205 is generated as an ideal by formula_35 and formula_207.
Several algebraic properties of formula_21 are inherited by formula_15:
Topological properties of the formal power series ring.
The metric space formula_222 is complete.
The ring formula_15 is compact if and only if "R" is finite. This follows from Tychonoff's theorem and the characterisation of the topology on formula_15 as a product topology.
Weierstrass preparation.
The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem.
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 formula_225:
Then one can show that
The last one being valid in the ring formula_231
For "K" a field, the ring formula_232 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 over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain. Let 
and suppose "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". Define:
This 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 formula_15 is the ("X")-adic topology and formula_15 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"("X"2−"X") and "f"((1−"X")−1 − 1) are all well defined for any formal power series formula_239
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
where "f" is a known power series with "f"(0) = 0, then the coefficients of "g" can be explicitly computed using the Lagrange inversion formula.
Generalizations.
Formal Laurent series.
The formal Laurent series over a ring formula_21 are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree. That is, they are the series that can be written as
for some integer , so that there are only finitely many negative with formula_244. (This is different from the classical Laurent series of complex analysis.) For a non-zero formal Laurent series, the minimal integer formula_10 such that formula_246 is called the "order" of formula_143 and is denoted formula_248 (The order of the zero series is formula_249.) 
Multiplication of such series can be defined. Indeed, similarly to the definition for formal power series, the coefficient of "Xk" of two series with respective sequences of coefficients formula_54 and formula_55 is
formula_252
This sum has only finitely many nonzero terms because of the assumed vanishing of coefficients at sufficiently negative indices.
The formal Laurent series form the ring of formal Laurent series over formula_21, denoted by formula_254. It is equal to the localization of formula_15 with respect to the set of positive powers of formula_35. If formula_136 is a field, then formula_258 is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain formula_138.
As with the ring formula_15 of formal power series, the ring formula_254 of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric
formula_262
One may define formal differentiation for formal Laurent series in the natural (term-by-term) way. Precisely, the formal derivative of the formal Laurent series formula_143 above is
formula_264
which is again a formal Laurent series. If formula_143 is a non-constant formal Laurent series and with coefficients in a field of characteristic 0, then one has
formula_266
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 formula_220 is a field of characteristic 0. Then the map
is a formula_220-derivation that satisfies
The latter shows that the coefficient of formula_272 in formula_143 is of particular interest; it is called "formal residue of formula_143" and denoted formula_275. The map
is formula_220-linear, and by the above observation one has an exact sequence
Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any formula_279
Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to formula_286. Property (iii): any formula_143 can be written in the form formula_288, with formula_289 and formula_290: then formula_291 formula_290 implies formula_144 is invertible in formula_294 whence formula_295 Property (iv): Since formula_296 we can write formula_297 with formula_298. Consequently, formula_299 and (iv) follows from (i) and (iii). Property (v) is clear from the definition.
The Lagrange inversion formula.
As mentioned above, any formal series formula_300 with "f"0 = 0 and "f"1 ≠ 0 has a composition inverse formula_301 The following relation between the coefficients of "gn" and "f"−"k" holds (""):
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. Noting formula_304, we can apply the rules of calculus above, crucially Rule (iv) substituting formula_305, to get:
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 formula_307-modules formula_308 where α is a complex exponent. As a consequence, if "f" and "g" are as above, with formula_309, we can relate the complex powers of "f" / "X" and "g" / "X": precisely, if α and β are non-zero complex numbers with negative integer sum, formula_310 then
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 "X"α 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 "X"α to a corresponding coefficient "c"α, and is denoted formula_312. The set of all such formal power series is denoted formula_313 and it is given a ring structure by defining
and
Topology.
The topology on formula_316 is such that a sequence of its elements converges only if for each monomial "X"α the corresponding coefficient stabilizes. If "I" is finite, then this the "J"-adic topology, where "J" is the ideal of formula_316 generated by all the indeterminates in "XI". This does not hold if "I" is infinite. For example, if formula_318 then the sequence formula_319 with formula_320 does not converge with respect to any "J"-adic topology on "R", but clearly for each monomial the corresponding coefficient stabilizes.
As remarked above, the topology on a repeated formal power series ring like formula_321 is usually chosen in such a way that it becomes isomorphic as a topological ring to formula_322
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 formula_323 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 "x"1, ..., "xr" are elements of "I", then there is a "unique" map formula_324 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 "X"α 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 "X"α to a corresponding coefficient "c"α, and is denoted formula_325. The set of all such formal power series is denoted "R"«"XI"», and it is given a ring structure by defining addition pointwise
and multiplication by
where · denotes concatenation of words. These formal power series over "R" form the Magnus ring over "R".
On a semiring.
Given an alphabet formula_328 and a semiring formula_104. The formal power series over formula_104 supported on the language formula_331 is denoted by formula_332. It consists of all mappings formula_333, where formula_331 is the free monoid generated by the non-empty set formula_328.
The elements of formula_332 can be written as formal sums
where formula_338 denotes the value of formula_339 at the word formula_340. The elements formula_341 are called the coefficients of formula_339.
For formula_343 the support of formula_339 is the set
A series where every coefficient is either formula_346 or formula_347 is called the characteristic series of its support.
The subset of formula_332 consisting of all series with a finite support is denoted by formula_349 and called polynomials.
For formula_350 and formula_351, the sum formula_352 is defined by 
The (Cauchy) product formula_354 is defined by
The Hadamard product formula_356 is defined by
And the products by a scalar formula_358 and formula_359 by
With these operations formula_362 and formula_363 are semirings, where formula_364 is the empty word in formula_331.
These formal power series are used to model the behavior of weighted automata, in theoretical computer science, when the coefficients formula_338 of the series are taken to be the weight of a path with label formula_367 in the automata.
Replacing the index set by an ordered abelian group.
Suppose formula_368 is an ordered abelian group, meaning an abelian group with a total ordering formula_369 respecting the group's addition, so that formula_370. Let I be a well-ordered subset of formula_368, meaning I contains no infinite descending chain. Consider the set consisting of
for all such I, with formula_373 in a commutative ring formula_21, where we assume that for any index set, if all of the formula_373 are zero then the sum is zero. Then formula_376 is the ring of formal power series on formula_368; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same. Sometimes the notation formula_378 is used to denote formula_376.
Various properties of formula_21 transfer to formula_376. If formula_21 is a field, then so is formula_376. If formula_21 is an ordered field, we can order formula_376 by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set I associated to a non-zero coefficient. Finally if formula_368 is a divisible group and formula_21 is a real closed field, then formula_376 is a real closed field, and if formula_21 is algebraically closed, then so is formula_376.
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.