Editing Formal power series (section)

== Operations on formal power series ==
One can perform algebraic operations on power series to generate new power series.<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3 =Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor1-first=Daniel |editor1-last=Zwillinger |editor2-first=Victor Hugo |editor2-last=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=0.313 |page=18}} (Several previous editions as well.)</ref><ref name=formalpowerseries>{{cite journal | first=Ivan | last=Niven | author-link=Ivan Niven | title=Formal Power Series | journal=[[American Mathematical Monthly]] | volume=76 | issue=8 | date=October 1969 | pages=871–889 | doi=10.1080/00029890.1969.12000359}}</ref>

===Power series raised to powers===
For any [[natural number]] {{math|''n''}}, the {{mvar|n}}th power of a formal power series {{mvar|S}} is defined recursively by
<math display="block">\begin{align}S^1&=S\\
S^n&=S\cdot S^{n-1}\quad\text{for } n>1.\end{align}</math>
If {{math|''a''<sub>0</sub>}} is invertible in the ring of coefficients, one can prove that in the expansion
<math display="block"> \Big( \sum_{k=0}^\infty a_k X^k \Big)^{n} = \sum_{m=0}^\infty c_m X^m,</math>
the coefficients are given by <math>c_0 = a_0^n</math> and
<math display="block">
c_m = \frac{1}{m a_0} \sum_{k=1}^m (kn - m+k) a_{k} c_{m-k}
</math>
for <math>m \geq 1</math> if {{mvar|m}} is invertible in the ring of coefficients.<ref>{{cite arXiv |last=Finkel |first=Hal |title=The differential transformation method and Miller's recurrence |date=2010-07-13 |class=math.CA |eprint=1007.2178}}</ref><ref name="Gould">{{Cite journal |last=Gould |first=H. W. |date=1974 |title=Coefficient Identities for Powers of Taylor and Dirichlet Series |url=https://www.jstor.org/stable/2318904 |journal=The American Mathematical Monthly |volume=81 |issue=1 |pages=3–14 |doi=10.2307/2318904 |jstor=2318904 |issn=0002-9890}}</ref><ref>{{Cite journal |last=Zeilberger |first=Doron |year=1995 |title=The J.C.P. miller recurrence for exponentiating a polynomial, and its q-analog. |url=https://doi.org/10.1080/10236199508808006 |journal=Journal of Difference Equations and Applications |volume=1 |issue=1 |pages=57–60 |doi=10.1080/10236199508808006 |via=Taylor & Francis Online}}</ref>{{Efn|The formula is often attributed to [[J.C.P. Miller]], but it has a long history of rediscovery, dating back to least Euler's discovery in 1748.<ref name="Gould" />}}
In the case of formal power series with complex coefficients, its complex powers are well defined for series {{math|''f''}} with constant term equal to {{math|1}}. In this case, <math>f^{\alpha}</math> can be defined either by composition with the [[binomial series]] {{math|(1 + ''x'')<sup>''α''</sup>}}, or by composition with the exponential and the logarithmic series, <math>f^{\alpha} = \exp(\alpha\log(f)),</math> or as the solution of the differential equation (in terms of series) <math> f(f^{\alpha})' = \alpha f^{\alpha} f'</math> with constant term {{math|1}}; the three definitions are equivalent. The [[exponent rules]] <math>(f^\alpha)^\beta = f^{\alpha\beta}</math> and <math>f^\alpha g^\alpha = (fg)^\alpha</math> easily follow for formal power series {{math|''f'', ''g''}}.

=== Multiplicative inverse ===
The series

:<math>A = \sum_{n=0}^\infty a_n X^n \in R[[X]]</math>

is invertible in <math>R[[X]]</math> if and only if its constant coefficient <math>a_0</math> is invertible in <math>R</math>. This condition is necessary, for the following reason: if we suppose that <math>A</math> has an inverse <math>B = b_0 + b_1 x + \cdots</math> then the [[constant term]] <math>a_0b_0</math> of <math>A \cdot B</math> 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 <math>B</math> via the explicit recursive formula

:<math>\begin{align}
b_0 &= \frac{1}{a_0},\\
b_n &= -\frac{1}{a_0} \sum_{i=1}^n a_i b_{n-i}, \ \ \ n \geq 1.
\end{align}</math>

An important special case is that the [[geometric series]] formula is valid in <math>R[[X]]</math>:

:<math>(1 - X)^{-1} = \sum_{n=0}^\infty X^n.</math>

If <math>R=K</math> 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 <math>X</math>. This means that <math>K[[X]]</math> is a [[discrete valuation ring]] with uniformizing parameter <math>X</math>.

===Division===
The computation of a quotient <math>f/g=h</math>

:<math> \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, </math>

assuming the denominator is invertible (that is, <math>a_0</math> is invertible in the ring of scalars), can be performed as a product <math>f</math> and the inverse of <math>g</math>, or directly equating the coefficients in <math>f=gh</math>:

:<math>c_n = \frac{1}{a_0}\left(b_n - \sum_{k=1}^n a_k c_{n-k}\right).</math>

=== Extracting coefficients ===
The coefficient extraction operator applied to a formal power series

:<math>f(X) = \sum_{n=0}^\infty a_n X^n </math>

in ''X'' is written

:<math> \left[ X^m \right] f(X) </math>

and extracts the coefficient of ''X<sup>m</sup>'', so that

:<math> \left[ X^m \right] f(X) = \left[ X^m \right] \sum_{n=0}^\infty a_n X^n = a_m.</math>

=== Composition ===
Given two formal power series

:<math>f(X) = \sum_{n=1}^\infty a_n X^n = a_1 X + a_2 X^2 + \cdots</math>
:<math>g(X) = \sum_{n=0}^\infty b_n X^n = b_0 + b_1 X + b_2 X^2 + \cdots</math>
such that  <math>a_0=0,</math>
one may form the ''composition''

:<math>g(f(X)) = \sum_{n=0}^\infty b_n (f(X))^n = \sum_{n=0}^\infty c_n X^n,</math>

where the coefficients ''c''<sub>''n''</sub> are determined by "expanding out" the powers of ''f''(''X''):

:<math>c_n:=\sum_{k\in\N, |j|=n} b_k a_{j_1} a_{j_2} \cdots a_{j_k}.</math>

Here the sum is extended over all (''k'', ''j'') with <math>k\in\N</math> and <math>j\in\N_+^k</math> with <math>|j|:=j_1+\cdots+j_k=n.</math>

Since <math>a_0=0,</math> one  must have <math>k\le n</math> and <math>j_i\le n</math> for every <math>i. </math> This implies that the above sum is finite and that the coefficient <math>c_n</math> is the coefficient of <math>X^n</math> in the polynomial <math>g_n(f_n(X))</math>, where <math>f_n</math> and <math>g_n</math> are the polynomials obtained by truncating the series at <math>x^n,</math> that is, by removing all terms involving a power of <math>X</math>  higher than <math>n.</math>

A more explicit description of these coefficients is provided by [[Faà di Bruno's formula#Formal power series version|Faà di Bruno's formula]], at least in the case where the coefficient ring is a field of [[Characteristic (algebra)|characteristic 0]].

Composition is only valid when <math>f(X)</math> has ''no constant term'', so that each <math>c_n</math> depends on only a finite number of coefficients of <math>f(X)</math> and <math>g(X)</math>. In other words, the series for <math>g(f(X))</math> converges in the [[Completion (ring theory)#Krull topology|topology]] of <math>R[[X]]</math>.

==== Example ====
Assume that the ring <math>R</math> has characteristic 0 and the nonzero integers are invertible in <math>R</math>. If one denotes by <math>\exp(X)</math> the formal power series

:<math>\exp(X) = 1 + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \frac{X^4}{4!} + \cdots,</math>

then the equality

:<math>\exp(\exp(X) - 1) = 1 + X + X^2 + \frac{5X^3}6 + \frac{5X^4}8 + \cdots</math>

makes perfect sense as a formal power series, since the constant coefficient of <math>\exp(X) - 1</math> is zero.

=== Composition inverse ===
Whenever a formal series

:<math>f(X)=\sum_k f_k X^k \in R[[X]]</math>

has ''f''<sub>0</sub> = 0 and ''f''<sub>1</sub> being an invertible element of ''R'', there exists a series

:<math>g(X)=\sum_k g_k X^k</math>

that is the [[composition inverse]] of <math>f</math>, meaning that composing <math>f</math> with <math>g</math> gives the series representing the [[identity function]] <math>x = 0 + 1x + 0x^2+ 0x^3+\cdots</math>. The coefficients of <math>g</math> 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 [[#The Lagrange inversion formula|Lagrange inversion formula]] (discussed below) provides a powerful tool to compute the coefficients of ''g'', as well as the coefficients of the (multiplicative) powers of ''g''.

=== Formal differentiation ===
Given a formal power series

:<math>f = \sum_{n\geq 0} a_n X^n \in R[[X]],</math>

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

:<math> Df = f' = \sum_{n \geq 1} a_n n X^{n-1}.</math>

The symbol ''D'' is called the '''formal differentiation operator'''. This definition simply mimics term-by-term differentiation of a polynomial.

This operation is ''R''-[[linear operator|linear]]:

:<math>D(af + bg) = a \cdot Df + b \cdot Dg</math>

for any ''a'', ''b'' in ''R'' and any ''f'', ''g'' in <math>R[[X]].</math> Additionally, the formal derivative has many of the properties of the usual [[derivative]] of calculus. For example, the [[product rule]] is valid:

:<math>D(fg) \ =\ f \cdot (Dg) + (Df) \cdot g,</math>

and the [[chain rule]] works as well:

:<math>D(f\circ g ) = ( Df\circ g ) \cdot Dg,</math>

whenever the appropriate compositions of series are defined (see above under [[#Composition of series|composition of series]]).

Thus, in these respects formal power series behave like [[Taylor series]]. Indeed, for the ''f'' defined above, we find that

:<math>(D^k f)(0) = k! a_k, </math>

where ''D''<sup>''k''</sup> denotes the ''k''th formal derivative (that is, the result of formally differentiating ''k'' times).

=== Formal antidifferentiation ===
If <math>R</math> is a ring with characteristic zero and the nonzero integers are invertible in <math>R</math>, then given a formal power series

:<math>f = \sum_{n\geq 0} a_n X^n \in R[[X]],</math>

we define its '''formal antiderivative''' or '''formal indefinite integral''' by

:<math> D^{-1} f = \int f\ dX = C + \sum_{n \geq 0} a_n \frac{X^{n+1}}{n+1}.</math>

for any constant <math>C \in R</math>.

This operation is ''R''-[[linear operator|linear]]:

:<math>D^{-1}(af + bg) = a \cdot D^{-1}f + b \cdot D^{-1}g</math>

for any ''a'', ''b'' in ''R'' and any ''f'', ''g'' in <math>R[[X]].</math> Additionally, the formal antiderivative has many of the properties of the usual [[antiderivative]] of calculus. For example, the formal antiderivative is the [[Inverse function#Left and right inverses|right inverse]] of the formal derivative:

:<math>D(D^{-1}(f)) = f</math>

for any <math>f \in R[[X]]</math>.