In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful, especially in combinatorics, for providing compact representations of sequences and multisets, and for finding closed formulas for recursively defined sequences; this is known as the method of generating functions.

Informal introduction

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A formal power series can be loosely thought of as a "polynomial 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. For example, consider the series

$A\; =\; 1\; -\; 3x\; +\; 5x^2\; -\; 7x^3\; +\; 9x^4\; -\; 11x^5\; +\; \backslash cdots.$

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 −3, 5, −7, 9, −11, .... In other words, a formal power series is just an object that records a sequence of coefficients.

Arithmetic on formal power series is carried out by simply pretending that the series are polynomials. For example, if

$B\; =\; 2x\; +\; 4x^3\; +\; 6x^5\; +\; \backslash cdots,$

then we add A and B term by term:

$A\; +\; B\; =\; 1\; -\; x\; +\; 5x^2\; -\; 3x^3\; +\; 9x^4\; -\; 5x^5\; +\; \backslash cdots.$

We can multiply formal power series, again just by treating them as polynomials:

$AB\; =\; 2x\; -\; 6x^2\; +\; 14x^3\; -\; 26x^4\; +\; 44x^5\; +\; \backslash cdots.$

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

$44x^5\; =\; (1\backslash times\; 6x^5)\; +\; (5x^2\; \backslash times\; 4x^3)\; +\; (9x^4\; \backslash times\; 2x).\backslash ,\backslash !$

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.

Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.

Formal development

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Two definitions of the formal power series ring

We start with a commutative ring R. We want to define the ring of formal power series over R in the variable X, denoted by RX; elements of this ring should be thought of as power series whose coefficients are elements of R.

Perhaps the most efficient definition of RX is as the completion of the polynomial ring Rwith respect to the I-adic topology determined by the ideal I of R^* as a dense subspace. This method determines the ring structure and topological structure simultaneously.

However, it is possible to describe RX more explicitly and with less algebraic machinery, giving the ring structure and topological structure separately, as follows.

Ring structure

We begin with the set R^N of all infinite sequences in R. We define addition of two such sequences by

$$

\left( a_n \right) + \left( b_n \right) = \left( a_n + b_n \right)

and multiplication by

$$

\left( a_n \right) \times \left( b_n \right) = \left( \sum_{k=0}^n a_k b_{n-k} \right).

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, R^N becomes a commutative ring with zero element (0, 0, 0, ...) and multiplicative identity (1, 0, 0,...).

If we identify the element a of R with the sequence (a, 0, 0, ...) and define X := (0, 1, 0, 0, ...), then using the above definitions of addition and multiplication, we find that every sequence with only finitely many nonzero terms can be written as the finite sum

$$

(a_0, a_1, a_2, \ldots, a_N, 0, 0, \ldots) = a_0 + a_1 X + \cdots + a_N X^N = \sum_{n=0}^N a_n X^n.

Topological structure

We would like to extend the above formula to a similar one for arbitrary sequences in R^N, that is, we would like

$$

(a_0, a_1, a_2, a_3, \ldots) = \sum_{n=0}^\infty a_n X^n \qquad (1) to hold. However, for the infinite sum on the right to make sense, we need a notion of convergence in R^N, which involves introducing a topology on R^N. There are several equivalent ways to define the appropriate topology.

• We may give R^N the product topology, where each copy of R is given the discrete topology.
• We may introduce a metric (or "distance function"). For sequences (a_n) and (b_n) in R^N, let us define

$d((a\_n),\; (b\_n))\; =\; 2^\{-k\},\backslash ,\backslash !$

where k is the smallest natural number such that a_k ≠ b_k; if there is no such k, then the two sequences are identical, so we set their distance to be zero.

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

All of these definitions of the topology amount to declaring that two sequences (a_n) and (b_n) are "close" if their first few terms agree; the more terms agree, the closer they are.

Now we can make sense of equation (1); the partial sums of the infinite sum certainly converge to the sequence on the left hand side. In fact, any rearrangement of the series converges to the same limit.

One must check that 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 RX.

Universal property

The ring RX may be characterized by the following universal property. 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 x is an element of I, then there is a unique Φ : RX → S with the following properties:

• Φ is an R-algebra homomorphism
• Φ is continuous
• Φ(X) = x.

Operations on formal power series

Inverting series

The series

$\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; X^n$

in RX is invertible in RX if and only if its constant coefficient a₀ is invertible in R. An important special case is that the geometric series formula is valid in RX:

$$

\left( 1 - X \right)^{-1} = \sum_{n \ge 0} X^n.

Composition of series

Given formal power series

$f(X)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; a\_n\; X^n\; =\; a\_1\; X\; +\; a\_2\; X^2\; +\; \backslash cdots$

and

$g(X)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; b\_n\; X^n\; =\; b\_0\; +\; b\_1\; X\; +\; b\_2\; X^2\; +\; \backslash cdots,$

one may form the composition

$g(f(X))\; =\; \backslash sum\_\{n=0\}^\backslash infty\; b\_n\; f(X)^n\; =\; \backslash sum\_\{n=0\}^\backslash infty\; c\_n\; X^n,$

where the coefficients c_n are determined by "expanding out" the powers of f(X). A more explicit description of these coefficients is provided by Faà di Bruno's formula.

The critical point here is that this operation is only valid when f(X) has no constant term, so that the series for g(f(X)) converges in the topology of RX. In other words, each c_n depends on only a finite number of coefficients of f(X) and g(X).

= Example

=

If we denote by exp(X) the formal power series

$\backslash exp(X)\; =\; 1\; +\; x\; +\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^4\}\{4!\}\; +\; \backslash cdots,$

then the expression

$\backslash exp(\backslash exp(X)\; -\; 1)\; =\; 1\; +\; x\; +\; x^2\; +\; \backslash frac\{5x^3\}6\; +\; \backslash frac\{5x^4\}8\; +\; \backslash cdots$

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

$\backslash exp(\backslash exp(X))\; =\; e\; \backslash exp(\backslash exp(X)\; -\; 1)\; =\; e\; +\; ex\; +\; ex^2\; +\; \backslash frac\{5ex^3\}6\; +\; \backslash cdots$

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

Formal differentiation of series

Given a formal power series

$f\; =\; \backslash sum\_\{n\backslash geq\; 0\}\; a\_n\; X^n$

in RX, we define its formal derivative, denoted Df, by

$$

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 \,\!

for any a, b in R and any f, g in RX. 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; \,\!

and the chain rule works as well:

$$

D(f(u)) = (Df)(u) Du, \,\!

whenever the appropriate compositions of series are defined (see above under composition of series).

In a sense, all formal power series are Taylor series. Indeed, for the f defined above, we find that

$$

(D^k f)(0) = k! a_k, \,\! where D^k denotes the kth formal derivative (that is, the result of formally differentiating k times).

Algebraic properties of the formal power series ring

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

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

The maximal ideals of RX all arise from those in R in the following manner: an ideal M of RX 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 RX:

• if R is a local ring, then so is RX
• if R is Noetherian, then so is RX; this is a version of the Hilbert basis theorem
• if R is an integral domain, then so is RX

If R = K is a field, then KX has several additional properties.

• KX is a discrete valuation ring.
• KX is a unique factorization domain.

Topological properties of the formal power series ring

The metric space (RX, d) is complete.

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

Applications

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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 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 QX:

$$

\sin(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n+1)!} X^{2n+1}

$$

\cos(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n)!} X^{2n}

Then one can show that

$$

\sin^2 + \cos^2 = 1

and

$$

D \sin = \cos

as well as

$$

\sin (X+Y) = \sin(X) \cos(Y) + \cos(X) \sin(Y)

(the latter being valid in the ring QX,Y).

In algebra, the ring KX₁, ..., X_r (where K is a field) is often used as the "standard, most general" complete local ring over K.

Interpreting formal power series as functions

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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. If f=∑a_n Xⁿ is an element of RX, 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) = \sum_{n\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 RX is the (X)-adic topology and RX 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∈RX.

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} = \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

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.

Generalizations

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Formal Laurent series

A formal Laurent series over R is defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree. That is, we consider series of the form

$$

f = \sum_{n \ge -M} a_n X^n where M is an integer which depends on f. We may add and multiply such series using the same formal rules as for formal power series; note that multiplication makes sense because we have only allowed finitely many negative index terms. Under these operations, these elements form the ring of formal Laurent series over R, denoted by R((X)). It is a topological ring, and its relationship to formal power series is analogous to the relationship between power series and Laurent series.

If R = K is a field, then K((X)) may also be obtained as the field of fractions of the integral domain KX.

One may define formal differentiation for formal Laurent series in a natural way (term-by-term). If R is a field, then in addition to the rules listed above under formal differentiation of series, the quotient rule will also be valid.

Power series in several variables

It is relatively straightforward to extend the above ideas to define a formal power series ring over R in r variables, denoted RX₁,...,X_r. Elements of this ring may be expressed uniquely in the form

$$

\sum_{\mathbf{n}\in\Bbb{N}^r} a_\mathbf{n} \mathbf{X^n} where now n = (n₁,...,n_r) ∈ N^r, and Xⁿ denotes the monomial X₁^n₁...X_r^n_r. This sum converges for any choice of the coefficients a_n∈R, and the order of summation is immaterial.

Definition

One possible definition of RX₁,...,X_r is to take the completion of the polynomial ring Rin r variables with respect to the I-adic topology, where I is the ideal of R[X₁,...,X_r generated by X₁,...,X_r. That is, I is the ideal consisting of polynomials with zero constant term.

Alternatively, one may proceed in a similar way to the more explicit discussion given above for the single-variable case, giving the ring structure first in terms of "multi-dimensional" sequences, and then defining the topology.

The topology on RX₁,...,X_r is the J-adic topology, where J is the ideal of RX₁,...,X_r generated by X₁,...,X_r. That is, J is the ideal consisting of series with zero constant term. Therefore, two series are considered "close" if their first few terms agree, where "first few" means terms whose total degree n₁ + ... + n_r is small.

Warning

Although RX₁, X₂ and RX₁X₂ are isomorphic as rings, they do not carry the same topology. For example, the sequence of elements

$f\_n\; =\; X\_1^n,\; \backslash quad\; n\; \backslash geq\; 1$

converges to zero in RX₁, X₂ as n → ∞; however, in the ring RX₁X₂, it does not converge, since the copy of RX₁ embedded in RX₁X₂ has been given the discrete topology.

Operations

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

• Addition is carried out term-by-term.
• Multiplication is carried out simply by "multiplying out" the series.
• 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 only if the constant term of g is zero.

In the case of the formal derivative, there are now r different partial derivative operators, which differentiate with respect to each of the r variables. They all commute with each other, as they do for continuously differentiable functions.

Universal property

In the several variables case, the universal property characterizing RX₁, ..., 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, ..., x_r are elements of I, then there is a unique Φ : RX₁, ..., 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.

Replacing the index set by an ordered abelian group

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

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• Bell series are used to study the properties of multiplicative arithmetic functions

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