Formal Power Series - Wikipedia, the Free Encyclopedia
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Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series ring 5 Applications 6 Interpreting formal power series as functions 7 Generalizations 7.1 Formal Laurent series 7.1.1 Formal residue 7.2 The Lagrange inversion formula 7.3 Power series in several variables 7.3.1 Topology 7.3.2 Operations 7.3.3 Universal property 7.4 Non-commuting variables 7.5 Replacing the index set by an ordered abelian group 8 Examples and related topics 9 Notes 1 of 16 10/11/2012 19:27 Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series 10 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 B A −1, provided that the inverse of A exists. For example, one can use the definition of multiplication above to verify the familiar formula 2 of 16 10/11/2012 19:27 Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series 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 Xn, and is written e.g. [Xn] A, so that [X2] A = 5 and [X5] A = −11. Other examples include and 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 R[[X]] abstractly as the completion of the polynomial ring R[X] equipped with a particular metric. This automatically gives R[[X]] 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 R[[X]] more explicitly, and define the ring structure and topological structure separately, as follows. Ring structure As a set, R[[X]] can be constructed as the set RN of all infinite sequences of elements of R, indexed by the natural numbers (taken to include 0). Designating a sequence whose term at index n is an 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, RN becomes a commutative ring with zero element (0, 0, 0, ...) and multiplicative identity (1, 0, 0,...). 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 R into R[[X]] by sending any (constant) a ∈ R to the sequence (a, 0, 0, ...) and designates the sequence (0, 1, 0, 0, ...) by X; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as 3 of 16 10/11/2012 19:27 Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series these are precisely the polynomials in X. Given this, it is quite natural and convenient to designate a general sequence an by 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 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 RN is defined and a topology on RN is constructed. There are several equivalent ways to define the desired topology. We may give RN the product topology, where each copy of R is given the discrete topology. We may give RN the I-adic topology, where I = (X) is the ideal generated by X, which consists of all sequences whose first term a0 is zero. The desired topology could also be derived from the following metric. The distance between distinct N sequences (an) and (bn) in R , is defined to be where k is the smallest natural number such that ak ≠ bk; the distance between two equal sequences is of course zero. Informally, two sequences (an) and (bn) 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 X 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 an, since inclusion of the term for i = n gives the last (and in fact only) change to the coefficient of Xn. 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 R and is denoted by R[[X]]. 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 X occurs in only finitely many terms. 4 of 16 10/11/2012 19:27 Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series 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 Xn.