Formal power series

In mathematics, formal power series are devices that make it possible to employ much of the mathematical analysis machinery of power series in settings that do not have natural notions of "convergence". They are also useful to compactly describe sequences and to find closed formulas for recursion defined sequences; this is known as the method of generating functions and will be illustrated below. We start with a commutative ring (algebra) ''R''. We want to define the ring of formal power series over R in the variable X, denoted by ''R''''X''; each element of this ring can be written in a unique way as an infinite sum of the form ∑_''n''≥0 ''a''_''n'' ''X''^''n'' where the coefficients ''a''_''n'' are elements of ''R''; any choice of coefficients ''a''_''n'' is allowed. After making an appropriate choice of topology, ''R''''X'' becomes a topological ring wherein these infinite sums are well-defined and convergent. The addition and multiplication of such sums follow the usual laws of power series. === Formal construction === Start with the set ''R''^N of all infinite sequences in ''R''. Define addition of two such sequences by :$\backslash left(\; a\_n\; \backslash right)\; +\; \backslash left(\; b\_n\; \backslash right)\; =\; \backslash left(\; a\_n\; +\; b\_n\; \backslash right)$ and multiplication by :$\backslash left(\; a\_n\; \backslash right)\; \backslash times\; \backslash left(\; b\_n\; \backslash right)\; =\; \backslash left(\; \backslash sum\_\{k=0\}^n\; a\_k\; b\_\{n-k\}\; \backslash right).$ This is called the Cauchy product of the two sequences of coefficients, and is a sort of discrete convolution. This turns ''R''^N into a commutative ring with multiplicative identity (1,0,0,...). We identify the element ''a'' of ''R'' with the sequence (''a'',0,0,...) and define ''X'' := (0,1,0,0,...). Then every element of ''R''^N of the form (''a''₀, ''a''₁, ''a''₂,...,''a''_''N'',0,0,...) can be written as the ''finite'' sum :$\backslash sum\_\{n=0\}^N\; a\_n\; X^n$ In order to extend this expansion to infinite series, we need some "distance" metric space ''d'' on ''R''^N. We define ''d'' ((''a''_''n'' ), (''b''_''n'' )) = 2^-''k'', where ''k'' is the smallest natural number such that ''a''_''k'' ≠ ''b''_''k'' (if there is no such ''k'', then the two sequences are the same and we define their distance to be zero). This is a metric which turns ''R''^N into a topological ring, and the equation :$\backslash left(\; a\_n\; \backslash right)\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; a\_n\; X^n$ can now be rigorously proven using the notion of convergence arising from ''d''; in fact, any rearrangement of the series converges to the same limit (mathematics). This topological ring is the ring of formal power series over ''R'' and is denoted by ''R''''X''. === Properties === ''R''''X'' is an associative algebra over ''R'' which contains the ring ''R''[''X''] of polynomials over ''R''; the polynomials correspond to the sequences which end in zeros. The geometric series formula is valid in ''R''''X'': :$\backslash left(\; 1\; -\; X\; \backslash right)^\{-1\}\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; X^n$ An element ∑ ''a''_''n'' ''X''^''n'' of ''R''''X'' is invertible in ''R''''X'' if and only if its constant coefficient ''a''₀ is invertible in ''R''. This implies that the Jacobson radical of ''R''''X'' is the ideal (ring theory) generated by ''X'' and the Jacobson radical of ''R''. The maximal ideals of ''R''''X'' all arise from those in ''R'' in the following manner: an ideal ''M'' of ''R''''X'' is maximal if and only if ''M'' ∩ ''R'' is a maximal ideal of ''R'' and ''M'' is generated as an ideal by ''X'' and ''M'' ∩ ''R''. Several algebraic properties of ''R'' are inherited by ''R''''X'': * if ''R'' is a local ring, then so is ''R''''X'' * if ''R'' is noetherian ring, then so is ''R''''X'' * if ''R'' is an integral domain, then so is ''R''''X'' * if ''R'' is a field (mathematics), then ''R''''X'' is a discrete valuation ring. The metric space (''R''''X'', ''d'') is completeness (topology). The topology on ''R''''X'' is equal to the product topology on ''R''^N where ''R'' is equipped with the discrete topology. It follows from Tychonoff's theorem that ''R''''X'' is compact if and only if ''R'' is finite. The topology on ''R''''X'' can also be seen as the I-adic topology, where ''I'' = (''X'') is the ideal generated by ''X'' (which consists of all formal power series whose zeroth coefficient is zero). If ''K''=''R'' is a field, we can consider the quotient field of the integral domain ''K''''X''; it is denoted by ''K''((''X'')). Its elements are formal Laurent series of the form :$f\; =\; \backslash sum\_\{n\; \backslash ge\; -M\}\; a\_n\; X^n$ where ''M'' is an integer which depends on the Laurent series ''f''. ''K''((''X'')) is a topological field. === Formal power series as functions === In mathematical analysis, every convergent power series defines a function (mathematics) with values in the real number or complex number numbers. Formal power series can also be interpreted as functions, but one has to be careful with the function domain and codomain. If ''f''=∑''a''_''n'' ''X''^''n'' 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 :$f(x)\; =\; \backslash sum\_\{n\backslash ge\; 0\}\; a\_n\; x^n$ This latter series is guaranteed to converge in ''S'' given the above assumptions on ''x''. Furthermore, we have :$(f+g)(x)\; =\; f(x)\; +\; g(x)$ and :$(fg)(x)\; =\; f(x)\; g(x)$ 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: ''f''(0), ''f''(''X''²-''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'': :$f^\{-1\}\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; a^\{-n-1\}\; (a-f)^n$ If the formal power series ''g'' with ''g''(0) = 0 is given implicitly by the equation :$f(g)\; =\; X$ where ''f'' is a known power series with ''f''(0) = 0, then the coefficients of ''g'' can be explicitly computed using the Lagrange inversion theorem. === Differentiating formal power series === If ''f'' = ∑ ''a''_n ''X''_''n'' is an element of ''R''''X'', we define its ''formal derivative'' using the operator ''D'' as :$Df\; =\; \backslash sum\_\{n\; \backslash ge\; 1\}\; a\_n\; n\; X^\{n-1\}$ This operation is ''R''-linear operator: :$D(af\; +\; bg)\; =\; a\; Df\; +\; b\; Dg$ for ''a'', ''b'' in ''R'' and ''f'', ''g'' in ''R''''X''. The formal derivative has many of the properties of the continuous derivative of calculus. For example, the product rule is valid: :$D(fg)\; =\; f(Dg)\; +\; (Df)\; g$ and the chain rule works as well: :$D(f(u))\; =\; (Df)(u)\; Du$ In a sense, all formal power series are Taylor series, because if ''f''=∑''a''_''n'' ''X''^''n'', then, writing ''D''_''k'' as the ''k''th formal derivative, we find that :$(D\_k\; f)(0)\; =\; k!\; a\_k.$ One can also define differentiation for formal Laurent series in a natural way, and then the quotient rule, in addition to the rules listed above, will also be valid. === Power series in several variables === The fastest way to define the ring ''R''''X''₁,...,''X''_''r'' of formal power series over ''R'' in ''r'' variables starts with the ring ''S'' = ''R''[''X''₁,...,''X''_''r''] of polynomials over ''R''. Let ''I'' be the ideal in ''S'' generated by ''X''₁,...,''X''_''r'', consider the I-adic topology on ''S'', and form its completeness (topology). This results in a complete topological ring containing ''S'' which is denoted by ''R''''X''₁,...,''X''_''r''. For n=(''n''₁,...,''n''_''r'')∈N^''r'', we write Xⁿ = ''X''₁^''n''₁...''X''_''r''^''n''_''r''. Then every element of ''R''''X''₁,...,''X''_''r'' can be written in a unique way as a sum :$\backslash sum\_\{\backslash mathbf\{n\}\backslash in\backslash Bbb\{N\}^r\}\; a\_\backslash mathbf\{n\}\; \backslash mathbf\{X^n\}$ These sums converge for any choice of the coefficients ''a''_n∈''R'' and the order in which the elements are added doesn't matter. If ''J'' is the ideal in ''R''''X''₁,...,''X''_''r'' generated by ''X''₁,...,''X''_''r'' (i.e. ''J'' consists of those power series with zero constant coefficients), then the topology on ''R''''X''₁,...,''X''_''r'' is the ''J''-adic topology. Since ''R''''X''₁ is a commutative ring, we can define its power series ring, say ''R''''X''₁''X''₂. This ring is naturally ring isomorphism to the ring ''R''''X''₁,''X''₂ just defined, but as topological rings the two are different. If ''K'' = ''R'' is a field, then ''K''''X''₁,...,''X''_''r'' is a unique factorization domain. Similar to the situation described above, we can "apply" power series in several variables to other power series with zero constant coefficients. It is also possible to define partial derivatives for formal power series in a straightforward way. Partial derivatives commute, as they do for continuously differentiable functions. === Uses === 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'': :$\backslash sin(X)\; :=\; \backslash sum\_\{n\; \backslash ge\; 0\}\; \backslash frac\{(-1)^n\}\; \{(2n+1)!\}\; X^\{2n+1\}$ :$\backslash cos(X)\; :=\; \backslash sum\_\{n\; \backslash ge\; 0\}\; \backslash frac\{(-1)^n\}\; \{(2n)!\}\; X^\{2n\}$ Then one can show that :$\backslash sin^2\; +\; \backslash cos^2\; =\; 1$ and :$D\; \backslash sin\; =\; \backslash cos$ as well as :$\backslash sin\; (X+Y)\; =\; \backslash sin(X)\; \backslash cos(Y)\; +\; \backslash cos(X)\; \backslash sin(Y)$ (the latter being valid in the ring Q''X'',''Y''). As an example of the method of generating functions which arises frequently in combinatorics, consider the problem of finding a closed formula for the Fibonacci numbers ''f''_''n'' defined by ''f''₀ = 0, ''f''₁ = 1, and ''f''_''n'' = ''f''_''n''−1 + ''f''_''n''−2 for ''n'' ≥ 2. We work in the ring R''X'' and define the power series :$f\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; f\_n\; X^n$ ''f'' is called the ''generating function'' for the sequence (''f''_''n''). The generating function for the sequence (''f''_''n''−1) is ''Xf'' and that of (''f''_''n''−2) is ''X''²''f''. From the recurrence relation, we therefore see that the power series ''Xf'' + ''X''²''f'' agrees with ''f'' except for the first two coefficients. Taking these into account, we find that :$f\; =\; Xf\; +\; X^2\; f\; +\; X$ (this is the crucial step; recurrence relations can almost always be translated into equations for the generating functions). Solving this equation for ''f'', we get :$f\; =\; \backslash frac\{X\}\; \{1\; -\; X\; -\; X^2\}$ The denominator can be factored using the golden ratio φ₁ = (1 + √5)/2 and φ₂ = (1 − √5)/2, and the technique of partial fraction decomposition yields :$\backslash frac\{1\; /\; \backslash sqrt\{5\}\}\; \{1-\backslash phi\_1\; X\}\; -\; \backslash frac\{1/\backslash sqrt\{5\}\}\; \{1-\; \backslash phi\_2\; X\}$ These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula :$f\_n\; =\; \backslash frac\{1\}\; \{\backslash sqrt\{5\}\}\; (\backslash phi\_1^n\; -\; \backslash phi\_2^n)$ In algebra, the ring ''K''''X''₁, ..., ''X''_''r'' (where ''K'' is a field) is often used as the "standard, most general" complete local ring over ''K''. === Universal property === The power series ring ''R''''X''₁, ..., ''X''_''r'' can be characterized by the following universal property: if ''S'' is a commutative associative algebra over ''R'', if ''I'' is an ideal in ''S'' such that the ''I''-adic topology on ''S'' is complete, and if ''x''₁, ..., ''x''_''r'' are elements of ''I'', then there is a ''unique'' Φ : ''R''''X''₁, ..., ''X''_''n'' -> ''S'' with the following properties: * Φ is an ''R''-algebra homomorphism * Φ is continuous * Φ(''X''_''i'') = ''x''_''i'' for ''i'' = 1, ..., ''r''. === Generalized formal power series === Suppose ''G'' is an ordered abelian group, meaning an abelian group with a total ordering "<" respecting the group's addition, so that ''a'' < ''b'' iff ''a'' + ''c'' < ''b'' + ''c'' for all ''c''. Let I be a well-ordered subset of ''G'', meaning I contains no infinite descending chain. Consider the set consisting of :$\backslash sum\_\{i\; \backslash in\; I\}\; a\_i\; X^i$ for all such I, with ''a''_''i'' in a commutative ring ''R'', where we assume that for any index set, if all of the ''a''_''i'' are zero then the sum is zero. Then ''R''((''G'')) is the ring of formal power series on ''G''; 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. Various properties of ''R'' transfer to ''R''((''G'')). If ''R'' is a field, then so is ''R''((''G'')). If ''R'' is an ordered field, we can order ''R''((''G'')) 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 ''G'' is a divisible group and ''R'' is a real closed field, then ''R''((''G'')) is a real closed field, and if ''R'' is algebraically closed, then so is ''R''((''G'')). 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 topics== * Bell series are used to study the properties of Multiplicative function Abstract algebra Ring theory Combinatorics Mathematical series

Formal power series

I've slightly improved (in my opinion) the sentence on ''k''!''a''_k in the section on differentiation. However, I'd probably prefer ''k''! to be regarded as a natural number, and to view this term as module multiplication of the additive group of ''R'' regarded as a module over the integers. I'm not sure how to state that concisely. How about : here ''na''_''k'' = ''a''_''k''+...+''a''_''k'' (''n'' summands) ? == Formal power series as functions == I think this article is quite well written, although I don't know if the introduction of the metric d() is standard and/or necessary for the notation $\backslash sum\; a\_kX^K$, which I was taught to be a mere notation for the sequence $(a\_k)$. I like this approach, but it may lead to confusion. Indeed, this metric is the only one introduced on this page, while the section "Formal power series as functions", starting with In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. clearly does not refer to this metric (for "convergent"), but to the topology in C (or in some complex function space), for the series of partial sums (of numbers or functions?) associated to the power series in the "obvious" way (but a precise define is not so immediate at all...). As it stands, this phrase is meaningless, because any power series is convergent according to what precedes, even if it has convergence radius of zero. So I think we should change it, so that this subsection becomes as "perfect" as the rest of the article. If the author of this article can find a more precise introduction for this subsection in the same style as what precedes, I would appreciate. (I don't know what is the most "gentle" way to go from a power series to the associated analytic function.)

User:MFH— User:MFH: User talk:MFH 12:53, 26 Apr 2005 (UTC)

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