In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).
A formal power series is a special kind of formal series, of the form
where the called coefficients, are numbers or, more generally, elements of some ring, and the are formal powers of the symbol that is called an indeterminate or, commonly, a variable. Hence, formal power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a formal power series may not represent a function of its variables. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.
A formal power series with coefficients in a ring is called a formal power series over The formal power series over a ring form a ring, commonly denoted by (It can be seen as the -adic completion of the polynomial ring in the same way as the -adic integers are the -adic completion of the ring of the integers.)
Formal powers series in several indeterminates are defined similarly by replacing the powers of a single indeterminate by monomials in several indeterminates.
Formal power series are widely used in combinatorics for representing sequences of integers as generating functions. In this context, a recurrence relation between the elements of a sequence may often be interpreted as a differential equation that the generating function satisfies. This allows using methods of complex analysis for combinatorial problems (see analytic combinatorics).
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 by the Cauchy–Hadamard theorem. 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.
Algebra 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 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 and called the ring of formal power series in the variable X over R.
Definition of the formal power series ring 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 structure 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 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 is defined and a topology on is constructed. There are several equivalent ways to define the desired topology.
We may give the product topology, where each copy of is given the discrete 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 (), 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 (in which case the product is nonzero) or infinitely many factors have no constant term (in which case the product is zero).
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 already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.
In the ring of formal power series , 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 (and its sum can be written as ); however
would be considered to be divergent, since every term affects the coefficient of . 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. With this topology, a sequence of elements of converges if the coefficient of each power of converges to a formal power series in , a weaker condition than stabilizing entirely. For instance, with this topology, in the second example given above, 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