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 x5{displaystyle x^{5}}x^{5} 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.




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 Power series raised to powers


    • 3.2 Inverting series


    • 3.3 Dividing series


    • 3.4 Extracting coefficients


    • 3.5 Composition of series


      • 3.5.1 Example




    • 3.6 Composition inverse


    • 3.7 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


    • 4.3 Weierstrass preparation




  • 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 On a semiring


    • 7.6 Replacing the index set by an ordered abelian group




  • 8 Examples and related topics


  • 9 Notes


  • 10 References


  • 11 Further reading





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


A=1−3X+5X2−7X3+9X4−11X5+⋯.{displaystyle A=1-3X+5X^{2}-7X^{3}+9X^{4}-11X^{5}+cdots .}A = 1 - 3X + 5X^2 - 7X^3 + 9X^4 - 11X^5 + cdots.

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


B=2X+4X3+6X5+⋯,{displaystyle B=2X+4X^{3}+6X^{5}+cdots ,}B = 2X + 4X^3 + 6X^5 + cdots,

then we add A and B term by term:


A+B=1−X+5X2−3X3+9X4−5X5+⋯.{displaystyle A+B=1-X+5X^{2}-3X^{3}+9X^{4}-5X^{5}+cdots .}A + B = 1 - X + 5X^2 - 3X^3 + 9X^4 - 5X^5 + cdots.

We can multiply formal power series, again just by treating them as polynomials (see in particular Cauchy product):


AB=2X−6X2+14X3−26X4+44X5+⋯.{displaystyle AB=2X-6X^{2}+14X^{3}-26X^{4}+44X^{5}+cdots .}AB = 2X - 6X^2 + 14X^3 - 26X^4 + 44X^5 + cdots.

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


44X5=(1×6X5)+(5X2×4X3)+(9X4×2X).{displaystyle 44X^{5}=(1times 6X^{5})+(5X^{2}times 4X^{3})+(9X^{4}times 2X).}44X^5 = (1times 6X^5) + (5X^2 times 4X^3) + (9X^4 times 2X).

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


11+X=1−X+X2−X3+X4−X5+⋯.{displaystyle {frac {1}{1+X}}=1-X+X^{2}-X^{3}+X^{4}-X^{5}+cdots .}frac{1}{1 + X} = 1 - X + X^2 - X^3 + X^4 - X^5 + cdots.

An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator [Xn]{displaystyle [X^{n}]}{displaystyle [X^{n}]} applied to a formal power series A{displaystyle A}A in one variable extracts the coefficient of the n{displaystyle n}nth power of the variable, so that [X2]A=5{displaystyle [X^{2}]A=5}{displaystyle [X^{2}]A=5} and [X5]A=−11{displaystyle [X^{5}]A=-11}{displaystyle [X^{5}]A=-11}. Other examples include


[X3](B)=4,[X2](X+3X2Y3+10Y6)=3Y3,[X2Y3](X+3X2Y3+10Y6)=3,[Xn](11+X)=(−1)n,[Xn](X(1−X)2)=n.{displaystyle {begin{aligned}left[X^{3}right](B)&=4,\left[X^{2}right](X+3X^{2}Y^{3}+10Y^{6})&=3Y^{3},\left[X^{2}Y^{3}right](X+3X^{2}Y^{3}+10Y^{6})&=3,\left[X^{n}right]left({frac {1}{1+X}}right)&=(-1)^{n},\left[X^{n}right]left({frac {X}{(1-X)^{2}}}right)&=n.end{aligned}}}{displaystyle {begin{aligned}left[X^{3}right](B)&=4,\left[X^{2}right](X+3X^{2}Y^{3}+10Y^{6})&=3Y^{3},\left[X^{2}Y^{3}right](X+3X^{2}Y^{3}+10Y^{6})&=3,\left[X^{n}right]left({frac {1}{1+X}}right)&=(-1)^{n},\left[X^{n}right]left({frac {X}{(1-X)^{2}}}right)&=n.end{aligned}}}

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]],{displaystyle R[[X]],}{displaystyle 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]]{displaystyle R[[X]]}R[[X]] abstractly as the completion of the polynomial ring R[X]{displaystyle R[X]}R[X] equipped with a particular metric. This automatically gives R[[X]]{displaystyle R[[X]]}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]]{displaystyle R[[X]]}R[[X]] more explicitly, and define the ring structure and topological structure separately, as follows.



Ring structure


As a set, R[[X]]{displaystyle R[[X]]}R[[X]] can be constructed as the set RN{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} of all infinite sequences of elements of R{displaystyle R}R, indexed by the natural numbers (taken to include 0). Designating a sequence whose term at index n{displaystyle n}n is an{displaystyle a_{n}}a_{n} by (an){displaystyle (a_{n})}(a_{n}), one defines addition of two such sequences by


(an)n∈N+(bn)n∈N=(an+bn)n∈N{displaystyle (a_{n})_{nin mathbb {N} }+(b_{n})_{nin mathbb {N} }=left(a_{n}+b_{n}right)_{nin mathbb {N} }}(a_n)_{ninN} + (b_n)_{ninN} = left( a_n + b_n right)_{ninN}

and multiplication by


(an)n∈N×(bn)n∈N=(∑k=0nakbn−k)n∈N.{displaystyle (a_{n})_{nin mathbb {N} }times (b_{n})_{nin mathbb {N} }=left(sum _{k=0}^{n}a_{k}b_{n-k}right)_{nin mathbb {N} }.}(a_n)_{ninN} times (b_n)_{ninN} = left( sum_{k=0}^n a_k b_{n-k} right)_{ninN}.

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{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} becomes a commutative ring with zero element (0,0,0,…){displaystyle (0,0,0,ldots )}{displaystyle (0,0,0,ldots )} and multiplicative identity (1,0,0,…){displaystyle (1,0,0,ldots )}{displaystyle (1,0,0,ldots )}.


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{displaystyle R}R into R[[X]]{displaystyle R[[X]]}R[[X]] by sending any (constant) a∈R{displaystyle ain R}a in R to the sequence (a,0,0,…){displaystyle (a,0,0,ldots )}{displaystyle (a,0,0,ldots )} and designates the sequence (0,1,0,0,…){displaystyle (0,1,0,0,ldots )}{displaystyle (0,1,0,0,ldots )} by X{displaystyle X}X; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as


(a0,a1,a2,…,an,0,0,…)=a0+a1X+⋯+anXn=∑i=0naiXi;{displaystyle (a_{0},a_{1},a_{2},ldots ,a_{n},0,0,ldots )=a_{0}+a_{1}X+cdots +a_{n}X^{n}=sum _{i=0}^{n}a_{i}X^{i};}(a_0, a_1, a_2, ldots, a_n, 0, 0, ldots) = a_0 + a_1 X + cdots + a_n X^n = sum_{i=0}^n a_i X^i;

these are precisely the polynomials in X{displaystyle X}X. Given this, it is quite natural and convenient to designate a general sequence (an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }}(a_n)_{ninN} by the formal expression ∑i∈NaiXi{displaystyle textstyle sum _{iin mathbb {N} }a_{i}X^{i}}textstylesum_{iinN}a_i X^i, 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


(∑i∈NaiXi)+(∑i∈NbiXi)=∑i∈N(ai+bi)Xi{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)+left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{iin mathbb {N} }(a_{i}+b_{i})X^{i}}{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)+left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{iin mathbb {N} }(a_{i}+b_{i})X^{i}}

and


(∑i∈NaiXi)×(∑i∈NbiXi)=∑n∈N(∑k=0nakbn−k)Xn.{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)times left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{nin mathbb {N} }left(sum _{k=0}^{n}a_{k}b_{n-k}right)X^{n}.}{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)times left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{nin mathbb {N} }left(sum _{k=0}^{n}a_{k}b_{n-k}right)X^{n}.}

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


(a0,a1,a2,a3,…)=∑i=0∞aiXi,(1){displaystyle (a_{0},a_{1},a_{2},a_{3},ldots )=sum _{i=0}^{infty }a_{i}X^{i},qquad (1)}(a_0, a_1, a_2, a_3, ldots) = sum_{i=0}^infty a_i X^i, qquad (1)

one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in RN{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} is defined and a topology on RN{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} is constructed. There are several equivalent ways to define the desired topology.


  • We may give RN{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} the product topology, where each copy of R{displaystyle R}R is given the discrete topology.

  • We may give RN{displaystyle R^{mathbb {N} }}{displaystyle R^{mathbb {N} }} the I-adic topology, where I=(X){displaystyle I=(X)}{displaystyle I=(X)} is the ideal generated by X{displaystyle X}X, which consists of all sequences whose first term a0{displaystyle a_{0}}a_{0} is zero.

  • The desired topology could also be derived from the following metric. The distance between distinct sequences (an),(bn)∈RN,{displaystyle (a_{n}),(b_{n})in R^{mathbb {N} },}{displaystyle (a_{n}),(b_{n})in R^{mathbb {N} },} is defined to be


d((an),(bn))=2−k,{displaystyle d((a_{n}),(b_{n}))=2^{-k},}{displaystyle d((a_{n}),(b_{n}))=2^{-k},}

where k{displaystyle k}k is the smallest natural number such that ak≠bk{displaystyle a_{k}neq b_{k}}{displaystyle a_{k}neq b_{k}}; the distance between two equal sequences is of course zero.


Informally, two sequences {an}{displaystyle {a_{n}}}{a_{n}} and {bn}{displaystyle {b_{n}}}{b_{n}} 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{displaystyle X}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{displaystyle a_{n}}a_{n}, since inclusion of the term for i=n{displaystyle i=n}i=n gives the last (and in fact only) change to the coefficient of Xn{displaystyle X^{n}}X^{n}. 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{displaystyle R}R and is denoted by R[[X]]{displaystyle R[[X]]}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{displaystyle X}X 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


(∑i∈NaiXi)×(∑i∈NbiXi)=∑i,j∈NaibjXi+j,{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)times left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{i,jin mathbb {N} }a_{i}b_{j}X^{i+j},}{displaystyle left(sum _{iin mathbb {N} }a_{i}X^{i}right)times left(sum _{iin mathbb {N} }b_{i}X^{i}right)=sum _{i,jin mathbb {N} }a_{i}b_{j}X^{i+j},}

since only finitely many terms on the right affect any fixed Xn{displaystyle X^{n}}X^{n}. 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


∑i=0∞aiXi{displaystyle sum _{i=0}^{infty }a_{i}X^{i}}{displaystyle sum _{i=0}^{infty }a_{i}X^{i}}

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 R{displaystyle R}R 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: Z[[X]][[Y]],{displaystyle mathbb {Z} [[X]][[Y]],}{displaystyle mathbb {Z} [[X]][[Y]],} then the topology of above construction only relates to the indeterminate Y{displaystyle Y}Y, since the topology that was put on Z[[X]]{displaystyle mathbb {Z} [[X]]}{displaystyle mathbb {Z} [[X]]} has been replaced by the discrete topology when defining the topology of the whole ring. So


∑i∈NXYi{displaystyle sum _{iin mathbb {N} }XY^{i}}sum_{iinN}XY^i

converges to the power series suggested, which can be written as X1−Y{displaystyle {tfrac {X}{1-Y}}}{displaystyle {tfrac {X}{1-Y}}}; however the summation


∑i∈NXiY{displaystyle sum _{iin mathbb {N} }X^{i}Y}sum_{iinN}X^iY

would be considered to be divergent, since every term affects the coefficient of Y{displaystyle Y}Y (which coefficient is itself a power series in X{displaystyle X}X). This asymmetry disappears if the power series ring in Y{displaystyle Y}Y is given the product topology where each copy of Z[[X]]{displaystyle mathbb {Z} [[X]]}{displaystyle mathbb {Z} [[X]]} 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 Z[[X]][[Y]]{displaystyle mathbb {Z} [[X]][[Y]]}{displaystyle mathbb {Z} [[X]][[Y]]} it then suffices that the coefficient of each power of Y{displaystyle Y}Y converges to a formal power series in X{displaystyle X}X, a weaker condition that stabilizing entirely; for instance in the second example given here the coefficient of Y{displaystyle Y}Yconverges to 11−X{displaystyle {tfrac {1}{1-X}}}{displaystyle {tfrac {1}{1-X}}}, so the whole summation converges to Y1−X{displaystyle {tfrac {Y}{1-X}}}{displaystyle {tfrac {Y}{1-X}}}.


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 Z[[X,Y]],{displaystyle mathbb {Z} [[X,Y]],}{displaystyle mathbb {Z} [[X,Y]],} and here a sequence converges if and only if the coefficient of every monomial XiYj{displaystyle X^{i}Y^{j}}{displaystyle X^{i}Y^{j}} stabilizes. This topology, which is also the I{displaystyle I}I-adic topology, where I=(X,Y){displaystyle I=(X,Y)}{displaystyle I=(X,Y)} is the ideal generated by X{displaystyle X}X and Y{displaystyle Y}Y, 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 R[[X]]{displaystyle mathbb {R} [[X]]}{displaystyle mathbb {R} [[X]]} the limit


limn→∞(1+Xn)n{displaystyle lim _{nto infty }left(1+{frac {X}{n}}right)^{n}}lim_{ntoinfty}left(1+frac{X}{n}right)^n

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


exp⁡(X)=∑n∈NXnn!.{displaystyle exp(X)=sum _{nin mathbb {N} }{frac {X^{n}}{n!}}.}{displaystyle exp(X)=sum _{nin mathbb {N} }{frac {X^{n}}{n!}}.}

This is because for i≥2{displaystyle igeq 2}igeq 2 the coefficient (ni)/ni{displaystyle {tbinom {n}{i}}/n^{i}}tbinom{n}{i}/n^i of Xi{displaystyle X^{i}}X^i does not stabilize as n→∞{displaystyle nto infty }nto infty . It does however converge in the usual topology of R{displaystyle mathbb {R} }mathbb {R} , and in fact to the coefficient 1i!{displaystyle {tfrac {1}{i!}}}{displaystyle {tfrac {1}{i!}}} of exp⁡(X){displaystyle exp(X)}{displaystyle exp(X)}. Therefore, if one would give R[[X]]{displaystyle mathbb {R} [[X]]}{displaystyle mathbb {R} [[X]]} the product topology of RN{displaystyle mathbb {R} ^{mathbb {N} }}{displaystyle mathbb {R} ^{mathbb {N} }} where the topology of R{displaystyle mathbb {R} }mathbb {R} is the usual topology rather than the discrete one, then the above limit would converge to exp⁡(X){displaystyle exp(X)}{displaystyle exp(X)}. 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 R[[X]]{displaystyle R[[X]]}R[[X]] may be characterized by the following universal property. If S{displaystyle S}S is a commutative associative algebra over R{displaystyle R}R, if I{displaystyle I}I is an ideal of S{displaystyle S}S such that the I{displaystyle I}I-adic topology on S{displaystyle S}S is complete, and if x{displaystyle x}x is an element of I{displaystyle I}I, then there is a unique Φ:R[[X]]→S{displaystyle Phi :R[[X]]to S}{displaystyle Phi :R[[X]]to S} with the following properties:



  • Φ{displaystyle Phi }Phi is an R{displaystyle R}R-algebra homomorphism


  • Φ{displaystyle Phi }Phi is continuous


  • Φ(X)=x{displaystyle Phi (X)=x}{displaystyle Phi (X)=x}.


Operations on formal power series


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 powers


If n is a natural number we have


(∑k=0∞akXk)n=∑m=0∞cmXm,{displaystyle left(sum _{k=0}^{infty }a_{k}X^{k}right)^{n}=sum _{m=0}^{infty }c_{m}X^{m},}{displaystyle left(sum _{k=0}^{infty }a_{k}X^{k}right)^{n}=sum _{m=0}^{infty }c_{m}X^{m},}

where:


c0=a0ncm=1ma0∑k=1m(kn−m+k)akcm−km≥1{displaystyle {begin{aligned}c_{0}&=a_{0}^{n}\c_{m}&={frac {1}{ma_{0}}}sum _{k=1}^{m}(kn-m+k)a_{k}c_{m-k}&&mgeq 1end{aligned}}}{displaystyle {begin{aligned}c_{0}&=a_{0}^{n}\c_{m}&={frac {1}{ma_{0}}}sum _{k=1}^{m}(kn-m+k)a_{k}c_{m-k}&&mgeq 1end{aligned}}}

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


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, fα{displaystyle f^{alpha }}{displaystyle f^{alpha }} can be defined either by composition with the binomial series (1+x)α, or by composition with the exponential and the logarithmic series, fα=exp⁡(αlog⁡(f)),{displaystyle f^{alpha }=exp(alpha log(f)),}{displaystyle f^{alpha }=exp(alpha log(f)),} or as the solution of the differential equation f(fα)′=αfαf′{displaystyle f(f^{alpha })'=alpha f^{alpha }f'}{displaystyle f(f^{alpha })'=alpha f^{alpha }f'} with constant term 1, the three definitions being equivalent. The rules of calculus (fα)β=fαβ{displaystyle (f^{alpha })^{beta }=f^{alpha beta }}{displaystyle (f^{alpha })^{beta }=f^{alpha beta }} and fαgα=(fg)α{displaystyle f^{alpha }g^{alpha }=(fg)^{alpha }}{displaystyle f^{alpha }g^{alpha }=(fg)^{alpha }} easily follow.



Inverting series


The series


A=∑n=0∞anXn∈R[[X]]{displaystyle A=sum _{n=0}^{infty }a_{n}X^{n}in R[[X]]}{displaystyle A=sum _{n=0}^{infty }a_{n}X^{n}in R[[X]]}

is invertible in R[[X]]{displaystyle R[[X]]}R[[X]] if and only if its constant coefficient a0{displaystyle a_{0}}a_{0} is invertible in R{displaystyle R}R. This condition is necessary, for the following reason: if we suppose that A{displaystyle A}A has an inverse B=b0+b1x+⋯{displaystyle B=b_{0}+b_{1}x+cdots }{displaystyle B=b_{0}+b_{1}x+cdots } then the constant term a0b0{displaystyle a_{0}b_{0}}a_{0}b_{0} of A⋅B{displaystyle Acdot B}Acdot B 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 B{displaystyle B}B via the explicit recursive formula


b0=1a0bn=−1a0∑i=1naibn−ifor n≥1.{displaystyle {begin{aligned}b_{0}&={frac {1}{a_{0}}}\b_{n}&=-{frac {1}{a_{0}}}sum _{i=1}^{n}a_{i}b_{n-i}&&{text{for }}ngeq 1.end{aligned}}}{displaystyle {begin{aligned}b_{0}&={frac {1}{a_{0}}}\b_{n}&=-{frac {1}{a_{0}}}sum _{i=1}^{n}a_{i}b_{n-i}&&{text{for }}ngeq 1.end{aligned}}}

An important special case is that the geometric series formula is valid in K[[X]]{displaystyle K[[X]]}{displaystyle K[[X]]}:


(1−X)−1=∑n=0∞Xn.{displaystyle (1-X)^{-1}=sum _{n=0}^{infty }X^{n}.}(1 - X)^{-1} = sum_{n=0}^infty X^n.

If R=K{displaystyle R=K}{displaystyle R=K} 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 X{displaystyle X}X. This says that K[[X]]{displaystyle K[[X]]}{displaystyle K[[X]]} is a discrete valuation ring with uniformizing parameter X{displaystyle X}X.



Dividing series


The computation of a quotient f/g=h{displaystyle f/g=h}{displaystyle f/g=h}


∑n=0∞bnXn∑n=0∞anXn=∑n=0∞cnXn,{displaystyle {frac {sum _{n=0}^{infty }b_{n}X^{n}}{sum _{n=0}^{infty }a_{n}X^{n}}}=sum _{n=0}^{infty }c_{n}X^{n},}{displaystyle {frac {sum _{n=0}^{infty }b_{n}X^{n}}{sum _{n=0}^{infty }a_{n}X^{n}}}=sum _{n=0}^{infty }c_{n}X^{n},}

assuming the denominator is invertible (that is, a0{displaystyle a_{0}}a_{0} is invertible in the ring of scalars), can be performed as a product f{displaystyle f}f and the inverse of g{displaystyle g}g, or directly equating the coefficients in f=gh{displaystyle f=gh}{displaystyle f=gh}:


cn=1a0(bn−∑k=1nakcn−k).{displaystyle c_{n}={frac {1}{a_{0}}}left(b_{n}-sum _{k=1}^{n}a_{k}c_{n-k}right).}c_n = frac{1}{a_0}left(b_n - sum_{k=1}^n a_k c_{n-k}right).


Extracting coefficients


The coefficient extraction operator applied to a formal power series


f(X)=∑n=0∞anXn{displaystyle f(X)=sum _{n=0}^{infty }a_{n}X^{n}}f(X) = sum_{n=0}^infty a_n X^n

in X is written


[Xm]f(X){displaystyle left[X^{m}right]f(X)} left[ X^m right] f(X)

and extracts the coefficient of Xm, so that


[Xm]f(X)=[Xm]∑n=0∞anXn=am.{displaystyle left[X^{m}right]f(X)=left[X^{m}right]sum _{n=0}^{infty }a_{n}X^{n}=a_{m}.} left[ X^m right] f(X) = left[ X^m right] sum_{n=0}^infty a_n X^n = a_m.


Composition of series


Given formal power series



f(X)=∑n=1∞anXn=a1X+a2X2+⋯{displaystyle f(X)=sum _{n=1}^{infty }a_{n}X^{n}=a_{1}X+a_{2}X^{2}+cdots }f(X) = sum_{n=1}^infty a_n X^n = a_1 X + a_2 X^2 + cdots

g(X)=∑n=0∞bnXn=b0+b1X+b2X2+⋯,{displaystyle g(X)=sum _{n=0}^{infty }b_{n}X^{n}=b_{0}+b_{1}X+b_{2}X^{2}+cdots ,}g(X) = sum_{n=0}^infty b_n X^n = b_0 + b_1 X + b_2 X^2 + cdots,


one may form the composition


g(f(X))=∑n=0∞bn(f(X))n=∑n=0∞cnXn,{displaystyle g(f(X))=sum _{n=0}^{infty }b_{n}(f(X))^{n}=sum _{n=0}^{infty }c_{n}X^{n},}g(f(X)) = sum_{n=0}^infty b_n (f(X))^n = sum_{n=0}^infty c_n X^n,

where the coefficients cn are determined by "expanding out" the powers of f(X):


cn:=∑k∈N,|j|=nbkaj1aj2⋯ajk.{displaystyle c_{n}:=sum _{kin mathbb {N} ,|j|=n}b_{k}a_{j_{1}}a_{j_{2}}cdots a_{j_{k}}.}{displaystyle c_{n}:=sum _{kin mathbb {N} ,|j|=n}b_{k}a_{j_{1}}a_{j_{2}}cdots a_{j_{k}}.}

Here the sum is extended over all (k, j) with k∈N{displaystyle kin mathbb {N} }{displaystyle kin mathbb {N} } and j∈N+k{displaystyle jin mathbb {N} _{+}^{k}}jinN_+^k with |j|:=j1+⋯+jk=n.{displaystyle |j|:=j_{1}+cdots +j_{k}=n.}|j|:=j_1+cdots+j_k=n.


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 f(X){displaystyle f(X)}f(X) has no constant term, so that each cn{displaystyle c_{n}}c_{n} depends on only a finite number of coefficients of f(X){displaystyle f(X)}f(X) and g(X){displaystyle g(X)}g(X). In other word the series for g(f(X)){displaystyle g(f(X))}{displaystyle g(f(X))} converges in the topology of R[[X]]{displaystyle R[[X]]}R[[X]].



Example


Assume that the ring R{displaystyle R}R has characteristic 0. If we denote by exp⁡(X){displaystyle exp(X)}{displaystyle exp(X)} the formal power series


exp⁡(X)=1+X+X22!+X33!+X44!+⋯,{displaystyle exp(X)=1+X+{frac {X^{2}}{2!}}+{frac {X^{3}}{3!}}+{frac {X^{4}}{4!}}+cdots ,}exp(X) = 1 + X + frac{X^2}{2!} + frac{X^3}{3!} + frac{X^4}{4!} + cdots,

then the expression


exp⁡(exp⁡(X)−1)=1+X+X2+5X36+5X48+⋯{displaystyle exp(exp(X)-1)=1+X+X^{2}+{frac {5X^{3}}{6}}+{frac {5X^{4}}{8}}+cdots }{displaystyle exp(exp(X)-1)=1+X+X^{2}+{frac {5X^{3}}{6}}+{frac {5X^{4}}{8}}+cdots }

makes perfect sense as a formal power series. However, the statement


exp⁡(exp⁡(X))=eexp⁡(exp⁡(X)−1)=e+eX+eX2+5eX36+⋯{displaystyle exp(exp(X))=eexp(exp(X)-1)=e+eX+eX^{2}+{frac {5eX^{3}}{6}}+cdots }exp(exp(X)) = e exp(exp(X) - 1) = e + eX + eX^2 + frac{5eX^3}{6} + cdots

is not a valid application of the composition operation for formal power series. Rather, it is confusing the notions of convergence in R[[X]]{displaystyle R[[X]]}R[[X]] and convergence in R{displaystyle R}R; indeed, the ring R{displaystyle R}R may not even contain any number e{displaystyle e}e with the appropriate properties.



Composition inverse


Whenever a formal series


f(X)=∑kfkXk∈R[[X]]{displaystyle f(X)=sum _{k}f_{k}X^{k}in R[[X]]}{displaystyle f(X)=sum _{k}f_{k}X^{k}in R[[X]]}

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


g(X)=∑kgkXk{displaystyle g(X)=sum _{k}g_{k}X^{k}}{displaystyle g(X)=sum _{k}g_{k}X^{k}}

that is the composition inverse of f{displaystyle f}f, meaning that composing f{displaystyle f}f with g{displaystyle g}g gives the series representing the identity function (whose first coefficient is 1 and all other coefficients are zero). The coefficients of g{displaystyle g}g 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 series


Given a formal power series


f=∑n≥0anXn∈R[[X]],{displaystyle f=sum _{ngeq 0}a_{n}X^{n}in R[[X]],}{displaystyle f=sum _{ngeq 0}a_{n}X^{n}in R[[X]],}

we define its formal derivative, denoted Df or f′, by


Df=∑n≥1annXn−1.{displaystyle Df=sum _{ngeq 1}a_{n}nX^{n-1}.} Df = sum_{n geq 1} a_n n X^{n-1}.

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:


D(af+bg)=a⋅Df+b⋅Dg{displaystyle D(af+bg)=acdot Df+bcdot Dg}D(af + bg) = a cdot Df + b cdot Dg

for any a, b in R and any f, g in R[[X]].{displaystyle R[[X]].}{displaystyle R[[X]].} Additionally, the formal derivative has many of the properties of the usual derivative of calculus. For example, the product rule is valid:


D(fg)=f⋅(Dg)+(Df)⋅g,{displaystyle D(fg)=fcdot (Dg)+(Df)cdot g,}D(fg) = f cdot (Dg) + (Df) cdot g,

and the chain rule works as well:


D(f∘g)=(Df∘g)⋅Dg,{displaystyle D(fcirc g)=(Dfcirc g)cdot Dg,}{displaystyle D(fcirc g)=(Dfcirc g)cdot Dg,}

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


(Dkf)(0)=k!ak,{displaystyle (D^{k}f)(0)=k!a_{k},}(D^k f)(0) = k! a_k,

where Dk denotes the kth formal derivative (that is, the result of formally differentiating k times).



Properties



Algebraic properties of the formal power series ring


R[[X]]{displaystyle R[[X]]}R[[X]] is an associative algebra over R{displaystyle R}R which contains the ring R[X]{displaystyle R[X]}R[X] of polynomials over R{displaystyle R}R; the polynomials correspond to the sequences which end in zeros.


The Jacobson radical of R[[X]]{displaystyle R[[X]]}R[[X]] is the ideal generated by X{displaystyle X}X and the Jacobson radical of R{displaystyle R}R; this is implied by the element invertibility criterion discussed above.


The maximal ideals of R[[X]]{displaystyle R[[X]]}R[[X]] all arise from those in R{displaystyle R}R in the following manner: an ideal M{displaystyle M}M of R[[X]]{displaystyle R[[X]]}R[[X]] is maximal if and only if M∩R{displaystyle Mcap R}{displaystyle Mcap R} is a maximal ideal of R{displaystyle R}R and M{displaystyle M}M is generated as an ideal by X{displaystyle X}X and M∩R{displaystyle Mcap R}{displaystyle Mcap R}.


Several algebraic properties of R{displaystyle R}R are inherited by R[[X]]{displaystyle R[[X]]}R[[X]]:


  • if R{displaystyle R}R is a local ring, then so is R[[X]]{displaystyle R[[X]]}R[[X]];

  • if R{displaystyle R}R is Noetherian, then so is R[[X]]{displaystyle R[[X]]}R[[X]]; this is a version of the Hilbert basis theorem;

  • if R{displaystyle R}R is an integral domain, then so is R[[X]]{displaystyle R[[X]]}R[[X]];

  • if K{displaystyle K}K is a field, then K[[X]]{displaystyle K[[X]]}{displaystyle K[[X]]} is a discrete valuation ring.


Topological properties of the formal power series ring


The metric space (R[[X]],d){displaystyle (R[[X]],d)}{displaystyle (R[[X]],d)} is complete.


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



Weierstrass preparation


Main article: Weierstrass preparation theorem § Formal power series in complete local rings

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 Q[[X]]{displaystyle mathbb {Q} [[X]]}{displaystyle mathbb {Q} [[X]]}:



sin⁡(X):=∑n≥0(−1)n(2n+1)!X2n+1{displaystyle sin(X):=sum _{ngeq 0}{frac {(-1)^{n}}{(2n+1)!}}X^{2n+1}}{displaystyle sin(X):=sum _{ngeq 0}{frac {(-1)^{n}}{(2n+1)!}}X^{2n+1}}

cos⁡(X):=∑n≥0(−1)n(2n)!X2n{displaystyle cos(X):=sum _{ngeq 0}{frac {(-1)^{n}}{(2n)!}}X^{2n}} cos(X) := sum_{n ge 0} frac{(-1)^n} {(2n)!} X^{2n}


Then one can show that



sin2⁡(X)+cos2⁡(X)=1,{displaystyle sin ^{2}(X)+cos ^{2}(X)=1,}sin^2(X) + cos^2(X) = 1,

∂∂Xsin⁡(X)=cos⁡(X),{displaystyle {frac {partial }{partial X}}sin(X)=cos(X),}frac{partial}{partial X} sin(X) = cos(X),

sin⁡(X+Y)=sin⁡(X)cos⁡(Y)+cos⁡(X)sin⁡(Y).{displaystyle sin(X+Y)=sin(X)cos(Y)+cos(X)sin(Y).}sin (X+Y) = sin(X) cos(Y) + cos(X) sin(Y).


The last one being valid in the ring Q[[X,Y]].{displaystyle mathbb {Q} [[X,Y]].}{displaystyle mathbb {Q} [[X,Y]].}


For K a field, the ring K[[X1,…,Xr]]{displaystyle K[[X_{1},ldots ,X_{r}]]}{displaystyle K[[X_{1},ldots ,X_{r}]]} 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 can also be interpreted as functions, but one has to be careful with the domain and codomain. Let


f=∑anXn∈R[[X]],{displaystyle f=sum a_{n}X^{n}in R[[X]],}{displaystyle f=sum a_{n}X^{n}in R[[X]],}

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:


f(x)=∑n≥0anxn.{displaystyle f(x)=sum _{ngeq 0}a_{n}x^{n}.}{displaystyle f(x)=sum _{ngeq 0}a_{n}x^{n}.}

This series is guaranteed to converge in S given the above assumptions on x. Furthermore, we have


(f+g)(x)=f(x)+g(x){displaystyle (f+g)(x)=f(x)+g(x)}{displaystyle (f+g)(x)=f(x)+g(x)}

and


(fg)(x)=f(x)g(x).{displaystyle (fg)(x)=f(x)g(x).}{displaystyle (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]]{displaystyle R[[X]]}R[[X]] is the (X)-adic topology and R[[X]]{displaystyle R[[X]]}R[[X]] 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(X2−X) and f((1−X)−1 − 1) are all well defined for any formal power series f∈R[[X]].{displaystyle fin R[[X]].}{displaystyle fin 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=∑n≥0a−n−1(a−f)n.{displaystyle f^{-1}=sum _{ngeq 0}a^{-n-1}(a-f)^{n}.}f^{-1} = sum_{n 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{displaystyle f(g)=X}{displaystyle 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 formula.



Generalizations



Formal Laurent series


A formal Laurent series over a ring R{displaystyle R}R 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


f=∑n∈ZanXn{displaystyle f=sum _{nin mathbb {Z} }a_{n}X^{n}}f = sum_{ninZ} a_n X^n

where an=0{displaystyle a_{n}=0}{displaystyle a_{n}=0} for all but finitely many negative indices n{displaystyle n}n. 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 {an}{displaystyle {a_{n}}}{a_{n}} and {bn}{displaystyle {b_{n}}}{b_{n}} is


∑i∈Zaibk−i,{displaystyle sum _{iin mathbb {Z} }a_{i}b_{k-i},}sum_{iinZ}a_ib_{k-i},

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


For a non-zero formal Laurent series, the minimal integer n{displaystyle n}n such that an≠0{displaystyle a_{n}neq 0}{displaystyle a_{n}neq 0} is called the order of f{displaystyle f}f, denoted ord⁡(f).{displaystyle operatorname {ord} (f).}{displaystyle operatorname {ord} (f).} (The order of the zero series is +∞{displaystyle +infty }+infty .) The formal Laurent series form the ring of formal Laurent series over R{displaystyle R}R, denoted by R((X)){displaystyle R((X))}{displaystyle R((X))}. It is equal to the localization of R[[X]]{displaystyle R[[X]]}R[[X]] with respect to the set of positive powers of X{displaystyle X}X. It is a topological ring with the metric:


d(f,g)=2−ord⁡(f−g).{displaystyle d(f,g)=2^{-operatorname {ord} (f-g)}.}{displaystyle d(f,g)=2^{-operatorname {ord} (f-g)}.}

If R=K{displaystyle R=K}{displaystyle R=K} is a field, then K((X)){displaystyle K((X))}{displaystyle K((X))} is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain K[[X]]{displaystyle K[[X]]}{displaystyle K[[X]]}.


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 f{displaystyle f}f above is


f′=Df=∑n∈ZnanXn−1{displaystyle f'=Df=sum _{nin mathbb {Z} }na_{n}X^{n-1}}f' = Df = sum_{ninZ} na_n X^{n-1}

which is again an element of K((X)){displaystyle K((X))}{displaystyle K((X))}. Notice that if f{displaystyle f}f is a non-constant formal Laurent series, and K is a field of characteristic 0, then one has


ord⁡(f′)=ord⁡(f)−1.{displaystyle operatorname {ord} (f')=operatorname {ord} (f)-1.}{displaystyle operatorname {ord} (f')=operatorname {ord} (f)-1.}

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 K{displaystyle K}K is a field of characteristic 0. Then the map


D:K((X))→K((X)){displaystyle Dcolon K((X))to K((X))}{displaystyle Dcolon K((X))to K((X))}

is a K{displaystyle K}K-derivation that satisfies



ker⁡D=K{displaystyle ker D=K}ker D=K

im⁡D={f∈K((X)):[X−1]f=0}.{displaystyle operatorname {im} D=left{fin K((X)):[X^{-1}]f=0right}.}{displaystyle operatorname {im} D=left{fin K((X)):[X^{-1}]f=0right}.}


The latter shows that the coefficient of X−1{displaystyle X^{-1}}{displaystyle X^{-1}} in f{displaystyle f}f is of particular interest; it is called formal residue of f{displaystyle f}f and denoted Res⁡(f){displaystyle operatorname {Res} (f)}{displaystyle operatorname {Res} (f)}. The map


Res:K((X))→K{displaystyle operatorname {Res} :K((X))to K}{displaystyle operatorname {Res} :K((X))to K}

is K{displaystyle K}K-linear, and by the above observation one has an exact sequence


0→K→K((X))→DK((X))→ResK→0.{displaystyle 0to Kto K((X)){xrightarrow {D}}K((X));{xrightarrow {operatorname {Res} }};Kto 0.}{displaystyle 0to Kto K((X)){xrightarrow {D}}K((X));{xrightarrow {operatorname {Res} }};Kto 0.}

Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any f,g∈K((X)){displaystyle f,gin K((X))}{displaystyle f,gin K((X))}



i. Res⁡(f′)=0;{displaystyle operatorname {Res} (f')=0;}{displaystyle operatorname {Res} (f')=0;}

ii. Res⁡(fg′)=−Res⁡(f′g);{displaystyle operatorname {Res} (fg')=-operatorname {Res} (f'g);}{displaystyle operatorname {Res} (fg')=-operatorname {Res} (f'g);}

iii. Res⁡(f′/f)=ord⁡(f),∀f≠0;{displaystyle operatorname {Res} (f'/f)=operatorname {ord} (f),qquad forall fneq 0;}{displaystyle operatorname {Res} (f'/f)=operatorname {ord} (f),qquad forall fneq 0;}

iv. Res⁡((f∘g)g′)=ord⁡(g)Res⁡(f),{displaystyle operatorname {Res} left((fcirc g)g'right)=operatorname {ord} (g)operatorname {Res} (f),}{displaystyle operatorname {Res} left((fcirc g)g'right)=operatorname {ord} (g)operatorname {Res} (f),} if ord⁡(g)>0;{displaystyle operatorname {ord} (g)>0;}{displaystyle operatorname {ord} (g)>0;}

v. [Xn]f(X)=Res⁡(X−n−1f(X)).{displaystyle [X^{n}]f(X)=operatorname {Res} left(X^{-n-1}f(X)right).}{displaystyle [X^{n}]f(X)=operatorname {Res} left(X^{-n-1}f(X)right).}


Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to (fg)′=f′g+fg′{displaystyle (fg)'=f'g+fg'}{displaystyle (fg)'=f'g+fg'}. Property (iii): any f{displaystyle f}f can be written in the form f=Xmg{displaystyle f=X^{m}g}{displaystyle f=X^{m}g}, with m=ord⁡(f){displaystyle m=operatorname {ord} (f)}{displaystyle m=operatorname {ord} (f)} and ord⁡(g)=0{displaystyle operatorname {ord} (g)=0}{displaystyle operatorname {ord} (g)=0}: then f′/f=mX−1+g′/g.{displaystyle f'/f=mX^{-1}+g'/g.}{displaystyle f'/f=mX^{-1}+g'/g.} ord⁡(g)=0{displaystyle operatorname {ord} (g)=0}{displaystyle operatorname {ord} (g)=0} implies g{displaystyle g}g is invertible in K[[X]]⊂im⁡(D)=ker⁡(Res),{displaystyle K[[X]]subset operatorname {im} (D)=ker(operatorname {Res} ),}{displaystyle K[[X]]subset operatorname {im} (D)=ker(operatorname {Res} ),} whence Res⁡(f′/f)=m.{displaystyle operatorname {Res} (f'/f)=m.}{displaystyle operatorname {Res} (f'/f)=m.} Property (iv): Since im⁡(D)=ker⁡(Res),{displaystyle operatorname {im} (D)=ker(operatorname {Res} ),}{displaystyle operatorname {im} (D)=ker(operatorname {Res} ),} we can write f=f−1X−1+F′,{displaystyle f=f_{-1}X^{-1}+F',}{displaystyle f=f_{-1}X^{-1}+F',} with F∈K[[X]]{displaystyle Fin K[[X]]}{displaystyle Fin K[[X]]}. Consequently, (f∘g)g′=f−1g−1g′+(F′∘g)g′=f−1g′/g+(F∘g)′{displaystyle (fcirc g)g'=f_{-1}g^{-1}g'+(F'circ g)g'=f_{-1}g'/g+(Fcirc g)'}{displaystyle (fcirc g)g'=f_{-1}g^{-1}g'+(F'circ g)g'=f_{-1}g'/g+(Fcirc g)'} and (iv) follows from (i) and (iii). Property (v) is clear from the definition.



The Lagrange inversion formula


Main article: Lagrange inversion theorem

As mentioned above, any formal series f∈K[[X]]{displaystyle fin K[[X]]}{displaystyle fin K[[X]]} with f0 = 0 and f1 ≠ 0 has a composition inverse g∈K[[X]].{displaystyle gin K[[X]].}{displaystyle gin K[[X]].} The following relation between the coefficients gn and f−k holds ("Lagrange inversion formula"):


k[Xk]gn=n[X−n]f−k.{displaystyle k[X^{k}]g^{n}=n[X^{-n}]f^{-k}.}{displaystyle k[X^{k}]g^{n}=n[X^{-n}]f^{-k}.}

In particular, for n = 1 and all k ≥ 1,


[Xk]g=1kRes⁡(f−k).{displaystyle [X^{k}]g={frac {1}{k}}operatorname {Res} left(f^{-k}right).}{displaystyle [X^{k}]g={frac {1}{k}}operatorname {Res} left(f^{-k}right).}

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting it here. Since ord⁡(f)=1{displaystyle operatorname {ord} (f)=1}{displaystyle operatorname {ord} (f)=1}, by the above rules of calculus,


k[Xk]gn=kRes⁡(gnX−k−1)=kRes⁡(Xnf−k−1f′)=−Res⁡(Xn(f−k)′)=Res⁡((Xn)′f−k)=nRes⁡(Xn−1f−k)=n[X−n]f−k.{displaystyle {begin{aligned}k[X^{k}]g^{n}&=koperatorname {Res} left(g^{n}X^{-k-1}right)=koperatorname {Res} left(X^{n}f^{-k-1}f,'right)=-operatorname {Res} left(X^{n}(f^{-k})^{'}right)\&=operatorname {Res} left(left(X^{n}right)'f^{-k}right)=noperatorname {Res} left(X^{n-1}f^{-k}right)=n[X^{-n}]f^{-k}.end{aligned}}}{displaystyle {begin{aligned}k[X^{k}]g^{n}&=koperatorname {Res} left(g^{n}X^{-k-1}right)=koperatorname {Res} left(X^{n}f^{-k-1}f,'right)=-operatorname {Res} left(X^{n}(f^{-k})^{'}right)\&=operatorname {Res} left(left(X^{n}right)'f^{-k}right)=noperatorname {Res} left(X^{n-1}f^{-k}right)=n[X^{-n}]f^{-k}.end{aligned}}}


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 C((X)){displaystyle mathbb {C} ((X))}{displaystyle mathbb {C} ((X))}-modules XαC((X)),{displaystyle X^{alpha }mathbb {C} ((X)),}{displaystyle X^{alpha }mathbb {C} ((X)),} where α is a complex exponent. As a consequence, if f and g are as above, with f1=g1=1{displaystyle f_{1}=g_{1}=1}f_1=g_1=1, 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, m=−α−β∈N,{displaystyle m=-alpha -beta in mathbb {N} ,}{displaystyle m=-alpha -beta in mathbb {N} ,} then


1α[Xm](fX)α=−1β[Xm](gX)β.{displaystyle {frac {1}{alpha }}[X^{m}]left({frac {f}{X}}right)^{alpha }=-{frac {1}{beta }}[X^{m}]left({frac {g}{X}}right)^{beta }.}{displaystyle {frac {1}{alpha }}[X^{m}]left({frac {f}{X}}right)^{alpha }=-{frac {1}{beta }}[X^{m}]left({frac {g}{X}}right)^{beta }.}

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 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 ∑αcαXα{displaystyle textstyle sum _{alpha }c_{alpha }X^{alpha }}textstylesum_alpha c_alpha X^alpha. The set of all such formal power series is denoted R[[XI]],{displaystyle R[[X_{I}]],}{displaystyle R[[X_{I}]],} and it is given a ring structure by defining


(∑αcαXα)+(∑αdαXα)=∑α(cα+dα)Xα{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)+left(sum _{alpha }d_{alpha }X^{alpha }right)=sum _{alpha }(c_{alpha }+d_{alpha })X^{alpha }}{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)+left(sum _{alpha }d_{alpha }X^{alpha }right)=sum _{alpha }(c_{alpha }+d_{alpha })X^{alpha }}

and


(∑αcαXα)×(∑βdβXβ)=∑α,βcαdβXα+β{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)times left(sum _{beta }d_{beta }X^{beta }right)=sum _{alpha ,beta }c_{alpha }d_{beta }X^{alpha +beta }}{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)times left(sum _{beta }d_{beta }X^{beta }right)=sum _{alpha ,beta }c_{alpha }d_{beta }X^{alpha +beta }}


Topology


The topology on R[[XI]]{displaystyle R[[X_{I}]]}{displaystyle R[[X_{I}]]} 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 R[[XI]]{displaystyle R[[X_{I}]]}{displaystyle R[[X_{I}]]} generated by all the indeterminates in XI. This does not hold if I is infinite. For example, if I=N,{displaystyle I=mathbb {N} ,}{displaystyle I=mathbb {N} ,} then the sequence (fn)n∈N{displaystyle (f_{n})_{nin mathbb {N} }}{displaystyle (f_{n})_{nin mathbb {N} }} with fn=Xn+Xn+1+Xn+2+⋯{displaystyle f_{n}=X_{n}+X_{n+1}+X_{n+2}+cdots }{displaystyle f_{n}=X_{n}+X_{n+1}+X_{n+2}+cdots } 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 R[[X]][[Y]]{displaystyle R[[X]][[Y]]}{displaystyle R[[X]][[Y]]} is usually chosen in such a way that it becomes isomorphic as a topological ring to R[[X,Y]].{displaystyle R[[X,Y]].}{displaystyle R[[X,Y]].}



Operations


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 property


In the several variables case, the universal property characterizing R[[X1,…,Xr]]{displaystyle R[[X_{1},ldots ,X_{r}]]}{displaystyle R[[X_{1},ldots ,X_{r}]]} 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 Φ:R[[X1,…,Xr]]→S{displaystyle Phi :R[[X_{1},ldots ,X_{r}]]to S}{displaystyle Phi :R[[X_{1},ldots ,X_{r}]]to S} with the following properties:


  • Φ is an R-algebra homomorphism

  • Φ is continuous

  • Φ(Xi) = xi for i = 1, ..., r.


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 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 ∑αcαXα{displaystyle textstyle sum _{alpha }c_{alpha }X^{alpha }}{displaystyle textstyle sum _{alpha }c_{alpha }X^{alpha }}. The set of all such formal power series is denoted R«XI», and it is given a ring structure by defining addition pointwise


(∑αcαXα)+(∑αdαXα)=∑α(cα+dα)Xα{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)+left(sum _{alpha }d_{alpha }X^{alpha }right)=sum _{alpha }(c_{alpha }+d_{alpha })X^{alpha }}left(sum_alpha c_alpha X^alpharight)+left(sum_alpha d_alpha X^alpharight)=sum_alpha(c_alpha+d_alpha)X^alpha

and multiplication by


(∑αcαXα)×(∑αdαXα)=∑α,βcαdβXα⋅Xβ{displaystyle left(sum _{alpha }c_{alpha }X^{alpha }right)times left(sum _{alpha }d_{alpha }X^{alpha }right)=sum _{alpha ,beta }c_{alpha }d_{beta }X^{alpha }cdot X^{beta }}left(sum_alpha c_alpha X^alpharight)timesleft(sum_alpha d_alpha X^alpharight)=sum_{alpha,beta} c_alpha d_beta X^{alpha} cdot X^{beta}

where · denotes concatenation of words. These formal power series over R form the Magnus ring over R.[3][4]



On a semiring


Given an alphabet Σ{displaystyle Sigma }Sigma and a semiring S{displaystyle S}S. The formal power series over S{displaystyle S}S supported on the language Σ∗{displaystyle Sigma ^{*}}Sigma ^{*} is denoted by S⟨⟨Σ∗⟩⟩{displaystyle Slangle langle Sigma ^{*}rangle rangle }{displaystyle Slangle langle Sigma ^{*}rangle rangle }. It consists of all mappings r:Σ∗→S{displaystyle r:Sigma ^{*}to S}{displaystyle r:Sigma ^{*}to S}, where Σ∗{displaystyle Sigma ^{*}}Sigma ^{*} is the free monoid generated by the non-empty set Σ{displaystyle Sigma }Sigma .


The elements of S⟨⟨Σ∗⟩⟩{displaystyle Slangle langle Sigma ^{*}rangle rangle }{displaystyle Slangle langle Sigma ^{*}rangle rangle } can be written as formal sums


r=∑w∈Σ∗(r,w)w.{displaystyle r=sum _{win Sigma ^{*}}(r,w)w.}{displaystyle r=sum _{win Sigma ^{*}}(r,w)w.}

where (r,w){displaystyle (r,w)}{displaystyle (r,w)} denotes the value of r{displaystyle r}r at the word w∈Σ∗{displaystyle win Sigma ^{*}}win Sigma ^{*}. The elements (r,w)∈S{displaystyle (r,w)in S}{displaystyle (r,w)in S} are called the coefficients of r{displaystyle r}r.


For r∈S⟨⟨Σ∗⟩⟩{displaystyle rin Slangle langle Sigma ^{*}rangle rangle }{displaystyle rin Slangle langle Sigma ^{*}rangle rangle } the support of r{displaystyle r}r is the set


supp⁡(r)={w∈Σ∗| (r,w)≠0}{displaystyle operatorname {supp} (r)={win Sigma ^{*}| (r,w)neq 0}}{displaystyle operatorname {supp} (r)={win Sigma ^{*}| (r,w)neq 0}}

A series where every coefficient is either 0{displaystyle 0}{displaystyle 0} or 1{displaystyle 1}1 is called the characteristic series of its support.


The subset of S⟨⟨Σ∗⟩⟩{displaystyle Slangle langle Sigma ^{*}rangle rangle }{displaystyle Slangle langle Sigma ^{*}rangle rangle } consisting of all series with a finite support is denoted by S⟨Σ∗⟩{displaystyle Slangle Sigma ^{*}rangle }{displaystyle Slangle Sigma ^{*}rangle } and called polynomials.


For r1,r2∈S⟨⟨Σ∗⟩⟩{displaystyle r_{1},r_{2}in Slangle langle Sigma ^{*}rangle rangle }{displaystyle r_{1},r_{2}in Slangle langle Sigma ^{*}rangle rangle } and s∈S{displaystyle sin S}sin S, the sum r1+r2{displaystyle r_{1}+r_{2}}{displaystyle r_{1}+r_{2}} is defined by


(r1+r2,w)=(r1,w)+(r2,w){displaystyle (r_{1}+r_{2},w)=(r_{1},w)+(r_{2},w)}{displaystyle (r_{1}+r_{2},w)=(r_{1},w)+(r_{2},w)}

The (Cauchy) product r1⋅r2{displaystyle r_{1}cdot r_{2}}{displaystyle r_{1}cdot r_{2}} is defined by


(r1⋅r2,w)=∑w1w2=w(r1,w1)(r2,w2){displaystyle (r_{1}cdot r_{2},w)=sum _{w_{1}w_{2}=w}(r_{1},w_{1})(r_{2},w_{2})}{displaystyle (r_{1}cdot r_{2},w)=sum _{w_{1}w_{2}=w}(r_{1},w_{1})(r_{2},w_{2})}

The Hadamard product r1⊙r2{displaystyle r_{1}odot r_{2}}{displaystyle r_{1}odot r_{2}} is defined by


(r1⊙r2,w)=(r1,w)(r2,w){displaystyle (r_{1}odot r_{2},w)=(r_{1},w)(r_{2},w)}{displaystyle (r_{1}odot r_{2},w)=(r_{1},w)(r_{2},w)}

And the products by a scalar sr1{displaystyle sr_{1}}{displaystyle sr_{1}} and r1s{displaystyle r_{1}s}{displaystyle r_{1}s} by



(sr1,w)=s(r1,w){displaystyle (sr_{1},w)=s(r_{1},w)}{displaystyle (sr_{1},w)=s(r_{1},w)} and (r1s,w)=(r1,w)s{displaystyle (r_{1}s,w)=(r_{1},w)s}{displaystyle (r_{1}s,w)=(r_{1},w)s}, respectively.

With these operations (S⟨⟨Σ∗⟩⟩,+,⋅,0,ε){displaystyle (Slangle langle Sigma ^{*}rangle rangle ,+,cdot ,0,varepsilon )}{displaystyle (Slangle langle Sigma ^{*}rangle rangle ,+,cdot ,0,varepsilon )} and (S⟨Σ∗⟩,+,⋅,0,ε){displaystyle (Slangle Sigma ^{*}rangle ,+,cdot ,0,varepsilon )}{displaystyle (Slangle Sigma ^{*}rangle ,+,cdot ,0,varepsilon )} are semirings, where ε{displaystyle varepsilon }varepsilon is the empty word in Σ∗{displaystyle Sigma ^{*}}Sigma ^{*}.


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



Replacing the index set by an ordered abelian group


Main article: Hahn series

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


∑i∈IaiXi{displaystyle sum _{iin I}a_{i}X^{i}}sum_{i in I} a_i X^i

for all such I, with ai{displaystyle a_{i}}a_{i} in a commutative ring R{displaystyle R}R, where we assume that for any index set, if all of the ai{displaystyle a_{i}}a_{i} are zero then the sum is zero. Then R((G)){displaystyle R((G))}{displaystyle R((G))} is the ring of formal power series on G{displaystyle G}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. Sometimes the notation [[RG]]{displaystyle [[R^{G}]]}{displaystyle [[R^{G}]]} is used to denote R((G)){displaystyle R((G))}{displaystyle R((G))}.[6]


Various properties of R{displaystyle R}R transfer to R((G)){displaystyle R((G))}{displaystyle R((G))}. If R{displaystyle R}R is a field, then so is R((G)){displaystyle R((G))}{displaystyle R((G))}. If R{displaystyle R}R is an ordered field, we can order R((G)){displaystyle R((G))}{displaystyle 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{displaystyle G}G is a divisible group and R{displaystyle R}R is a real closed field, then R((G)){displaystyle R((G))}{displaystyle R((G))} is a real closed field, and if R{displaystyle R}R is algebraically closed, then so is R((G)){displaystyle R((G))}{displaystyle 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 arithmetic functions


  • Formal groups are used to define an abstract group law using formal power series


  • Puiseux series are an extension of formal Laurent series, allowing fractional exponents

  • Rational series



Notes





  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. 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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em} (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.




References




  • Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007.


  • Nicolas Bourbaki: Algebra, IV, §4. Springer-Verlag 1988.



Further reading



  • 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




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