Formal power series ($R[[T]]$)

The formal power series ring $R[[T]]$ consists of all expressions ${\sum }_{n\ge 0}{a}_n{T}^n$ with $a_n\in R$. Addition is coefficientwise, while multiplication uses the Cauchy product; no analytic convergence is required.

For a formal power series over a field of characteristic zero, the displayed coefficientwise operation chooses constant coefficient zero. Its formal derivative is the original formal power series: differentiating the term with exponent $n+1$ cancels its denominator. Conversely, formally integrating a formal derivative removes the original constant term. These are algebraic identities; analytic convergence is not required.

For a formal power series $f(x)={\sum }_{n\ge 0}{a}_n{x}^n$ over a commutative ring $R$, a reciprocal is a series $g$ with $fg=1$. It exists exactly when $a_0$ is a unit. The constant coefficient forces $b_0=a_0^{-1}$, and the higher coefficients are determined successively byThus the reciprocal is unique and uses no analytic convergence assumption. For example, $(1-x{)}^{-1}=1+x+x^2+\cdots$ in $R[[x]]$. Newton inversion of a formal power series accelerates the computation when many coefficients are needed.

Over a field of characteristic zero, let $w(x)$ be the unique formal power series with constant coefficient zero satisfying $w=x\varphi (w)$, where $\varphi (0)\ne 0$. Lagrange inversion gives, for $n\ge 1$,Here the brackets denote coefficient extraction. The theorem obtains coefficients of an implicitly defined series without solving for it explicitly.

For $w=x(1+w{)}^2$ and $F(t)=t$, it gives $[x^n]w=\left(\genfrac{}{}{0ex}{}{2n}{n-1}\right)/n=\left(\genfrac{}{}{0ex}{}{2n}{n}\right)/(n+1)$, the Catalan numbers for $n\ge 1$. This is useful when a recursive counting construction naturally gives an implicit generating function.

If the constant coefficient of $A$ is invertible, then for every formal power series $B$ there is a unique quotient $Q$ satisfying $AQ=B$. Comparing coefficients determines $Q$ recursively; multiplying $B$ by the formal power series reciprocal of $A$ computes the same quotient.

For a formal power series $A$ with zero constant term over a characteristic-zero field, its exponential is $\exp A={\sum }_{k\ge 0}{A}^k/k!$. Every coefficient is a finite sum because $A^k$ starts in degree at least $k$. In finite characteristic, truncated computations require the factorial denominators that occur to be invertible.

To solve $\log G=A$ with $A(0)=0$, start with $G=1$ and update $G\leftarrow G(1+A-\log G)$ at successively doubled truncation precision. The correction removes the current logarithmic error to first order. This is Newton iteration applied to the formal power series exponential, using a formal power series logarithm.

A formal power series $f(x)$ over a field of characteristic zero is D-finite if it satisfies a nontrivial linear differential equation with polynomial coefficients:The polynomials are not all zero. Equivalently, the coefficients of $f$ satisfy a linear recurrence relation with polynomial functions of the index as coefficients.

For example, $e^x$ satisfies $f^{\prime }-f=0$, and every rational function regular at zero is D-finite. The class is closed under sums, products, and derivatives. This permits symbolic computation with many generating functions using finite equations rather than an infinite list of coefficients.

A q-series is a series in a parameter $q$, treated either as a formal power series or as an analytic series, often for $|q|<1$. Its coefficients can encode a sequence, while identities involving infinite products record choices of parts in integer partitions.

For example,where $p(n)$ counts integer partitions of $n$. Expanding each factor as a geometric series chooses the number of parts of size $m$. Formally, each coefficient uses only finitely many factors. This product is a basic link between combinatorics, number theory, and modular forms.

Here an inverse of a formal power series means its multiplicative inverse: $f(x{)}^{-1}f(x)=1$. The formal power series reciprocal exists exactly when the constant coefficient of $f$ is a unit in the coefficient ring. Thus $1-x$ is invertible in $\mathbb{Z}[[x]]$, while $2+x$ is not, though it is invertible in $\mathbb{Q}[[x]]$.

A compositional inverse solves $f(g(x))=x$, which is a different problem. For instance, the multiplicative inverse of $1-x$ is $1+x+x^2+\cdots$, whereas the compositional inverse of $x+x^2$ starts $x-x^2+2x^3-\cdots$. Stating which operation is inverted prevents a common ambiguity.

Suppose $f$ is a formal power series with invertible constant coefficient and $g$ satisfies $fg\equiv 1(\mathrm{mod}{x}^m)$. The updateis an inverse modulo $x^{2m}$, because $1-f{g}_{\mathrm{new}}=(1-fg{)}^2$ before truncation. Each update doubles the number of correct coefficients.

Start with the inverse of the constant coefficient and repeat, truncating all products to the required order. For $f=1-x$, the first approximations are $1$, $1+x$, and $1+x+x^2+x^3$. With a polynomial multiplication cost $M(n)$ satisfying the usual growth assumptions, computing the first $n$ coefficients takes $O(M(n))$ operations. The identity is algebraic and remains valid in characteristic two.

Over a commutative ring containing $\mathbb{Q}$, a formal power series $f=1+u$ with $u\in xR[[x]]$ has the formal logarithmEvery coefficient receives only finitely many contributions, since $u^k$ is divisible by $x^k$. No analytic convergence is involved. For example, $\log (1+x)=x-x^2/2+x^3/3-\cdots$.

The formal derivative satisfies $(\log f{)}^{\prime }=f^{\prime }/f$, and $\log (fg)=\log f+\log g$ for commuting series with constant coefficient one. This turns multiplicative identities for generating functions into additive ones. In positive characteristic, division by $k$ may fail, so this definition cannot be transferred unchanged.

For a commutative Noetherian ring, the ideals of lowest possible coefficients of an ideal in $R[[X]]$ form an ascending chain. Choose lifts of finite generating sets through the stabilization degree. Successive coefficient cancellation expresses any series in the ideal using these finitely many lifts. The multiplier series converge coefficientwise, so this generates the actual ideal without first assuming it is closed. Iterating gives the result for finitely many variables.

The formal power series ring $R={\mathbb{Z}}_p[[T]]$ is complete for two useful topologies. In the $p$-adic topology, convergence means that all coefficient errors are uniformly divisible by arbitrarily large powers of $p$. In the $(p,T)$-adic topology, only finitely many coefficients are controlled at each precision. The latter topology is compact, since $R/(p,T{)}^n$ is finite and $R$ is its inverse limit. The former is finer and is not compact: the series $T^n$ converge to zero in the latter but remain pairwise separated modulo $p$ in the former. Coleman norm contraction uses the former; compact interpolation arguments use the latter.

A formal power series with zero constant coefficient has a compositional inverse over $R$ exactly when its linear coefficient is a unit. Starting with $b_1=a_1^{-1}$, the degree-$k$ coefficient of $f(b_{1}T+\cdots +b_k{T}^k)$ is $a_1{b}_k$ plus an expression already determined by lower coefficients, so recursion gives a unique inverse. For a homomorphism of formal group laws, cancellation in the homomorphism identity proves that this inverse is also a formal-group homomorphism.

Let $R$ be a commutative unital ring and $f(T)=a_{1}T+a_2{T}^2+\cdots$ with $a_1$ a unit. Set $g(T)={\sum }_{n\ge 1}{b}_n{T}^n$. The coefficient of $T$ in $f(g(T))$ gives $b_1=a_1^{-1}$. At every degree $n>1$, the coefficient is $a_1{b}_n$ plus an expression involving only $b_1,\dots ,b_{n-1}$; choose $b_n$ to cancel that expression. This gives a unique right compositional inverse of a formal power series.

Similarly, the coefficient of $T^n$ in $h(f(T))$ is $a_1^n{h}_n$ plus previously fixed terms, so the same recursion gives a left inverse $h$. Composition of zero-constant-term formal power series is associative, since every coefficient involves finitely many terms. Therefore $h=h\circ (f\circ g)=(h\circ f)\circ g=g$, proving $f\circ g=g\circ f=T$.

If $(A,\mathfrak{m})$ is a local ring, then $B=A[[x]]$ is local with maximal ideal $I=\mathfrak{m}B+xB$ and residue field $A/\mathfrak{m}$. A series $f=\sum a_n{x}^n$ is a unit if and only if $a_0$ is a unit. Necessity follows by taking constant coefficients of an inverse; sufficiency follows from the recursion $b_0=a_0^{-1}$ and $b_n=-a_0^{-1}{\sum }_{i=1}^n{a}_i{b}_{n-i}$ for its inverse. Thus the nonunits are precisely series with $a_0\in \mathfrak{m}$, which is exactly $I$, since $f=a_0+xg$. The constant-coefficient map followed by reduction modulo $\mathfrak{m}$ has kernel $I$ and image $A/\mathfrak{m}$.

For a local ring $(A,\mathfrak{m})$ with residue field $k$, put $B=A[[x]]$ and $I=\mathfrak{m}B+xB$. There is a natural $k$-vector space isomorphism $I/I^2\cong (\mathfrak{m}/{\mathfrak{m}}^2)\oplus k$. The inverse sends $(\bar{a},\bar{b})$ to the class of $a+bx$. It is well-defined because changing $a$ by ${\mathfrak{m}}^2$ or $b$ by $\mathfrak{m}$ changes the result by $I^2$. It is surjective since every $f\in I$ is congruent modulo $x^{2}B\subseteq I^2$ to its constant and linear terms. As $I^2={\mathfrak{m}}^{2}B+x\mathfrak{m}B+x^{2}B$, those coefficients of an element of $I^2$ lie in ${\mathfrak{m}}^2$ and $\mathfrak{m}$, respectively. This proves injectivity. Thus adjoining one formal variable increases finite embedding dimension by one.

A formal power series ring is a reduced ring if and only if its coefficient ring is reduced. A nonzero constant nilpotent proves one direction. In the other direction, a nonzero series with first nonzero coefficient $a_r$ has first coefficient $a_r^m$ in its $m$th power; this is nonzero over a reduced ring.

For a vector space $M$ over a field $k$, its formal power series module is $M[[t]]={\underleftarrow{\lim }}_{n}M{\otimes }_{k}k[t]/(t^n)$. It is complete for the filtration by $t^{n}M[[t]]$. The ordinary tensor product $M{\otimes }_{k}k[[t]]$ embeds in it and consists of series whose coefficients span a finite-dimensional vector subspace; it equals $M[[t]]$ if $M$ is finite-dimensional. The inclusion is proper for infinite-dimensional $M$, as shown by a series with linearly independent coefficients.

A formal power series ring $R[[x]]$ is Noetherian if and only if $R$ is Noetherian. In the forward direction, use the quotient ring $R[[x]]/(x)\cong R$. In the reverse direction, for an ideal $I$ consider the ascending sequence of ideals of $R$ formed by the coefficient of $x^n$ in $I\cap x^{n}R[[x]]$. Stabilization and finite generation give a finite list of series that generate $I$ by successive coefficient cancellation.

For an ideal $H\subseteq R[[X]]$, let $A_n$ consist of coefficients of $X^n$ in members of $H$ with all lower coefficients zero. These are ideals of $R$ and $A_n\subseteq A_{n+1}$, by multiplication by $X$. If $R$ is Noetherian, this chain stabilizes. Choose series whose leading coefficients generate the finitely many distinct coefficient ideals, then cancel coefficients successively. The accumulated multipliers are formal power series, giving an ordinary finite ideal generating set for $H$, rather than merely a dense subideal.

This is the power-series specialization of a generating function.

The ordinary generating function of a sequence $(a_n{)}_{n\ge 0}$ is the formal power series $A(z)={\sum }_{n\ge 0}{a}_n{z}^n$. Algebraic identities for $A$ encode recurrences and counting constructions for the coefficients.