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; 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 is the fifth term in the sequence. In combinatorics, formal power series provide representations of numerical sequences and of multisets, and for instance allow concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring. Formal power series can be created from Taylor polynomials using formal moduli.

IntroductionEdit

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   applied to a formal power series   in one variable extracts the coefficient of the  th power of the variable, so that   and  . 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 seriesEdit

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   and called the ring of formal power series in the variable X over R.

Definition of the formal power series ringEdit

One can characterize   abstractly as the completion of the polynomial ring   equipped with a particular metric. This automatically gives   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   more explicitly, and define the ring structure and topological structure separately, as follows.

Ring structureEdit

As a set,   can be constructed as the set   of all infinite sequences of elements of  , indexed by the natural numbers (taken to include 0). Designating a sequence whose term at index   is   by  , 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,   becomes a commutative ring with zero element   and multiplicative identity  .

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   into   by sending any (constant)   to the sequence   and designates the sequence   by  ; 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  . Given this, it is quite natural and convenient to designate a general sequence   by the formal expression  , 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 structureEdit

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   is defined and a topology on   is constructed. There are several equivalent ways to define the desired topology.

  • We may give   the I-adic topology, where   is the ideal generated by  , which consists of all sequences whose first term   is zero.
  • The desired topology could also be derived from the following metric. The distance between distinct sequences   is defined to be
 
where   is the smallest natural number such that  ; the distance between two equal sequences is of course zero.

Informally, two sequences   and   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   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 (1), regardless of the values  , since inclusion of the term for   gives the last (and in fact only) change to the coefficient of  . 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   and is denoted by  . 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   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  . 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 topologiesEdit

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   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:  ; then the topology of above construction only relates to the indeterminate  , since the topology that was put on   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  ; however the summation

 

would be considered to be divergent, since every term affects the coefficient of   (which coefficient is itself a power series in  ). This asymmetry disappears if the power series ring in   is given the product topology where each copy of   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   it then suffices that the coefficient of each power of   converges to a formal power series in  , a weaker condition than stabilizing entirely; for instance in the second example given here the coefficient of  converges to  , so the whole summation converges to  .

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   and here a sequence converges if and only if the coefficient of every monomial   stabilizes. This topology, which is also the  -adic topology, where   is the ideal generated by   and  , 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   the limit

 

does not exist, so in particular it does not converge to

 

This is because for   the coefficient   of   does not stabilize as  . It does however converge in the usual topology of  , and in fact to the coefficient   of  . Therefore, if one would give   the product topology of   where the topology of   is the usual topology rather than the discrete one, then the above limit would converge to  . 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 propertyEdit

The ring   may be characterized by the following universal property. If   is a commutative associative algebra over  , if   is an ideal of   such that the  -adic topology on   is complete, and if   is an element of  , then there is a unique   with the following properties:

  •   is an  -algebra homomorphism
  •   is continuous
  •  .

Operations on formal power seriesEdit

One can perform algebraic operations on power series to generate new power series.[1][2] Besides the ring structure operations defined above, we have the following.

Power series raised to powersEdit

For any natural number n we have

 

where

 

(This formula can only be used if m and a0 are invertible in the ring of coefficients.)

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,   can be defined either by composition with the binomial series (1+x)α, or by composition with the exponential and the logarithmic series,   or as the solution of the differential equation   with constant term 1, the three definitions being equivalent. The rules of calculus   and   easily follow.

Inverting seriesEdit

The series

 

is invertible in   if and only if its constant coefficient   is invertible in  . This condition is necessary, for the following reason: if we suppose that   has an inverse   then the constant term   of   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   via the explicit recursive formula

 

An important special case is that the geometric series formula is valid in  :

 

If   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  . This means that   is a discrete valuation ring with uniformizing parameter  .

Dividing seriesEdit

The computation of a quotient  

 

assuming the denominator is invertible (that is,   is invertible in the ring of scalars), can be performed as a product   and the inverse of  , or directly equating the coefficients in  :

 

Extracting coefficientsEdit

The coefficient extraction operator applied to a formal power series

 

in X is written

 

and extracts the coefficient of Xm, so that

 

Composition of seriesEdit

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   and   with  

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   has no constant term, so that each   depends on only a finite number of coefficients of   and  . In other words, the series for   converges in the topology of  .

ExampleEdit

Assume that the ring   has characteristic 0 and the nonzero integers are invertible in  . If we denote by   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   and convergence in  ; indeed, the ring   may not even contain any number   with the appropriate properties.

Composition inverseEdit

Whenever a formal series

 

has f0 = 0 and f1 being an invertible element of R, there exists a series

 

that is the composition inverse of  , meaning that composing   with   gives the series representing the identity function (whose first coefficient is 1 and all other coefficients are zero). The coefficients of   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 provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.

Formal differentiation of seriesEdit

Given a formal power series

 

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   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).

PropertiesEdit

Algebraic properties of the formal power series ringEdit

  is an associative algebra over   which contains the ring   of polynomials over  ; the polynomials correspond to the sequences which end in zeros.

The Jacobson radical of   is the ideal generated by   and the Jacobson radical of  ; this is implied by the element invertibility criterion discussed above.

The maximal ideals of   all arise from those in   in the following manner: an ideal   of   is maximal if and only if   is a maximal ideal of   and   is generated as an ideal by   and  .

Several algebraic properties of   are inherited by  :

  • if   is a local ring, then so is  ,
  • if   is an integral domain, then so is  ,

Topological properties of the formal power series ringEdit

The metric space   is complete.

The ring   is compact if and only if R is finite. This follows from Tychonoff's theorem and the characterisation of the topology on   as a product topology.

Weierstrass preparationEdit

The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem.

ApplicationsEdit

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  :

 
 

Then one can show that

 
 
 

The last one being valid in the ring  

For K a field, the ring   is often used as the "standard, most general" complete local ring over K in algebra.

Interpreting formal power series as functionsEdit

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. 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   is the (X)-adic topology and   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(X2X) and f((1−X)−1 − 1) are all well defined for any formal power series  

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.

GeneralizationsEdit

Formal Laurent seriesEdit

A formal Laurent series over a ring   is defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree (this is different from the classical Laurent series), that is series of the form

 

where   for all but finitely many negative indices  . 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   and   is

 

which sum is effectively finite because of the assumed vanishing of coefficients at sufficiently negative indices, and which sum zero for sufficiently negative   for the same reason.

For a non-zero formal Laurent series, the minimal integer   such that   is called the order of  , denoted   (The order of the zero series is  .) The formal Laurent series form the ring of formal Laurent series over  , denoted by  . It is equal to the localization of   with respect to the set of positive powers of  . It is a topological ring with the metric:

 

If   is a field, then   is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain  .

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   above is

 

which is again an element of  . Notice that if   is a non-constant formal Laurent series, and K is a field of characteristic 0, then one has

 

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 residueEdit

Assume that   is a field of characteristic 0. Then the map

 

is a  -derivation that satisfies

 
 

The latter shows that the coefficient of   in   is of particular interest; it is called formal residue of   and denoted  . The map

 

is  -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  

i.  
ii.  
iii.  
iv.   if  
v.  

Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to  . Property (iii): any   can be written in the form  , with   and  : then     implies   is invertible in   whence   Property (iv): Since   we can write   with  . Consequently,   and (iv) follows from (i) and (iii). Property (v) is clear from the definition.

The Lagrange inversion formulaEdit

As mentioned above, any formal series   with f0 = 0 and f1 ≠ 0 has a composition inverse   The following relation between the coefficients gn and fk 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  , by the above rules of calculus,

 

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  -modules   where α is a complex exponent. As a consequence, if f and g are as above, with  , 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,   then

 

For instance, this way one finds the power series for complex powers of the Lambert function.

Power series in several variablesEdit

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 iI, 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  . The set of all such formal power series is denoted   and it is given a ring structure by defining

 

and

 

TopologyEdit

The topology on   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   generated by all the indeterminates in XI. This does not hold if I is infinite. For example, if   then the sequence   with   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   is usually chosen in such a way that it becomes isomorphic as a topological ring to  

OperationsEdit

All of the operations defined for series in one variable may be extended to the several variables case.

  • A series is invertible if and only if its constant term is invertible in R.
  • The composition f(g(X)) of two series f and g is defined if f is a series in a single indeterminate, and the constant term of g is zero. For a series f in several indeterminates a form of "composition" can similarly be defined, with as many separate series in the place of g as there are indeterminates.

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 propertyEdit

In the several variables case, the universal property characterizing   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 map   with the following properties:

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

Non-commuting variablesEdit

The several variable case can be further generalised by taking non-commuting variables Xi for iI, 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  . 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.[3][4]

On a semiringEdit

Given an alphabet   and a semiring  . The formal power series over   supported on the language   is denoted by  . It consists of all mappings  , where   is the free monoid generated by the non-empty set  .

The elements of   can be written as formal sums

 

where   denotes the value of   at the word  . The elements   are called the coefficients of  .

For   the support of   is the set

 

A series where every coefficient is either   or   is called the characteristic series of its support.

The subset of   consisting of all series with a finite support is denoted by   and called polynomials.

For   and  , the sum   is defined by

 

The (Cauchy) product   is defined by

 

The Hadamard product   is defined by

 

And the products by a scalar   and   by

  and  , respectively.

With these operations   and   are semirings, where   is the empty word in  .

These formal power series are used to model the behavior of weighted automata, in theoretical computer science, when the coefficients   of the series are taken to be the weight of a path with label   in the automata.[5]

Replacing the index set by an ordered abelian groupEdit

Suppose   is an ordered abelian group, meaning an abelian group with a total ordering   respecting the group's addition, so that   if and only if   for all  . Let I be a well-ordered subset of  , meaning I contains no infinite descending chain. Consider the set consisting of

 

for all such I, with   in a commutative ring  , where we assume that for any index set, if all of the   are zero then the sum is zero. Then   is the ring of formal power series on  ; 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   is used to denote  .[6]

Various properties of   transfer to  . If   is a field, then so is  . If   is an ordered field, we can order   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   is a divisible group and   is a real closed field, then   is a real closed field, and if   is algebraically closed, then so is  .

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.

Examples and related topicsEdit

NotesEdit

  1. ^ Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "0.313". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 18. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
  2. ^ Niven, Ivan (October 1969). "Formal Power Series". American Mathematical Monthly. 76 (8): 871–889. doi:10.1080/00029890.1969.12000359.
  3. ^ Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 167. ISBN 978-3-540-63003-6. Zbl 0819.11044.
  4. ^ Moran, Siegfried (1983). The Mathematical Theory of Knots and Braids: An Introduction. North-Holland Mathematics Studies. 82. Elsevier. p. 211. ISBN 978-0-444-86714-8. Zbl 0528.57001.
  5. ^ Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, p. 12
  6. ^ Shamseddine, Khodr; Berz, Martin (2010). "Analysis on the Levi-Civita Field: A Brief Overview" (PDF). Contemporary Mathematics. 508: 215–237.

ReferencesEdit

Further readingEdit

  • W. Kuich. Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, Chapter 9, pages 609–677. Springer, Berlin, 1997, ISBN 3-540-60420-0
  • Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1