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Questions tagged [formal-power-series]

This tag is for questions relating to "formal power series" which can be considered either as an extension of the polynomial to a possibly infinite number of terms or as a power series in which the variable is not assigned any value.

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Connection Between Derivations of Finite and Infinite Binomial Expansion

At first when learning the binomial expansion you learn it in the case of working as a shortcut to multiplying out brackets - anti-factorising if you will. In these cases what you are expanding takes ...
Ardavan Hamisi's user avatar
8 votes
1 answer
289 views

Seeking a Purely Formal Power Series Solution

Seeking a Purely Formal Power Series Solution When I read about generating functions, I encountered the following problem: Suppose that the set of nonnegative integers is partitioned into a finite ...
Math_fun2006's user avatar
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Can Zeckendorf's theorem be proven using generating functions?

First, I state Zeckendorf's theorem. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum ...
Math_fun2006's user avatar
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Compatibility - Topological modules contra vector spaces

So Tréves in his book on topological vector spaces shows that a filter $\mathcal{F}$ on a $\textit{vector space}$ $E$ is the filter of neighbourhoods of zero compatible with the linear structure if ...
undefined's user avatar
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1 answer
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Product Principle for Generating Functions?

Product Principle for Generating Functions I am inquiring about the Product Principle for Generating Functions as applied in combinatorial counting problems. First, let me state the principle: ...
Math_fun2006's user avatar
0 votes
1 answer
23 views

Fubini for formal power series

Let $A_{k,n} \in \mathbb{C}[[X]]$ such that $\lim _{k+n \rightarrow \infty} A_{k, n}=\mathbf{0}$. Prove that $$ \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} A_{k, n}=\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} ...
Math_fun2006's user avatar
0 votes
1 answer
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Formal power $ \sum_{k=1}^{\infty} A_k = \sum_{k=1}^{\infty} A_{\pi(k)}. $

Consider the sequence of elements $A_1, A_2, \cdots \in \mathbb{C}[[X]]$ satisfying $\lim A_n = \mathbf{0}$ and $\pi: \mathbb{N} \rightarrow \mathbb{N}$ being a bijection. Then, $$ \sum_{k=1}^{\infty} ...
Math_fun2006's user avatar
2 votes
1 answer
63 views

Counting irreducible Permutations

Let $x:=(x_1,...,x_n)$ be a permutation of $\{1,...,n\}$. We say, $x$ is irreducible iff $\{x_1,...,x_m\}\neq\{1,...,m\}$ for $1\leq m \leq n-1$. Let $g(n)$ be the number of irreduible permutations of ...
NTc5's user avatar
  • 609
5 votes
1 answer
113 views

If $f(g(x))=x$, does $g(f(x))=x$ hold?

If $f(x)$ and $g(x)$ are formal power series, i.e. $$f(x)=\sum_{n\ge 0} a_n x^n, g(x)=\sum_{n\ge 0} b_n x^n,$$ and $$f(g(x))=x,$$ can it be proved that always have $$g(f(x))=x?$$ It seems intuitive, ...
athos's user avatar
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2 votes
0 answers
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A generalization of integration by parts

Many years ago, I came up with a short generalization of integration by parts that was definitely known, but I could never find a reference for it. I was considering throwing it on arxiv, but before I ...
Terence C's user avatar
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Building the theoretical foundation for generating functions - formal power series

I have read several documents on generating functions. I would like to inquire about two issues: Among the materials I have read, some mention generating functions constructed from formal power ...
Math_fun2006's user avatar
0 votes
1 answer
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$\forall a,b \in K[[x]]: a | b \implies v(a) \leq v(b)$ where $v$ is the valuation

The valuation of a power series is the smallest index $i$ of the coefficients $a_i$ so that $a_i \ne 0$. Since $a|b$ there exists a $c\in K[[x]]$ so that $$\sum_{n=0}^\infty a_nx^n\sum_{n=0}^\infty ...
Xaver Wallenstein's user avatar
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1 answer
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find square root of $x^2+x^3 $ in formal power series $k[[x,y]]$

I am trying to show that the polynomial $y-x^2-x^3$ is reducible in the formal power series ring $k[[x,y]]$. I am attempting the question by finding a polynomial in $k[[x,y]]$ which is the square root ...
Siddharth Prakash's user avatar
2 votes
2 answers
85 views

Compute the power series of $\frac{2-6x+x^2}{1-3x}$

I'm struggling with the following: Suppose that we have a formal power series $\sum_{n \geq 0} a_nx^n$ which is equal to the fraction $\frac{2-6x+x^2}{1-3x}.$ I want to find an explicit formula for $...
DrTokus1998's user avatar
2 votes
0 answers
46 views

Convergence of $f(x)= \sum_{n=0}^{\infty} x^{2^n}$ [duplicate]

If we define $f(x)= \sum_{m=0}^{\infty} x^{2^m} = \lim_{n\to\infty}f_n(x)= \lim_{n\to\infty}\sum_{m=0}^{n} x^{2^m}$. It's easy to see that $|{f(x)}|$ converge in the interval $(-1,1)$. (We can easily ...
Student_Number_249812341's user avatar
3 votes
1 answer
83 views

Are modules for the ring of formal power series representations of a certain object?

Most of my exposure to modules has been over the various forms of category algebras, such as group algebras, path algebras, and incidence algebras. As such, I don't have a lot of exposure to modules ...
tox123's user avatar
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If $A\to B$ is an integral ring morphism, so is $A[[x]]\to B[[x]]$?

(Notation $R[[x]]$ means the ring of formal power series over $R$, a commutative ring with unity.) Observations: Since $A\to B$ integral implies $A[x]\to B[x]$ integral, $B[x]\subset B[[x]]$ is ...
Elías Guisado Villalgordo's user avatar
1 vote
0 answers
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Are there "brute force" method for Dong's lemma?

Dong's lemma: if $A(z),B(z),C(z)\in \operatorname{End}(V)[[z^{\pm 1}]]$ pairwise local, then $:\mathrel{A(z)B(z)}:$ and $C(z)$ are local. I'm trying the following approach for proving the lemma: We ...
Peter Wu's user avatar
  • 915
1 vote
1 answer
87 views

Why is $\wedge ^2 E[p] \cong \mu_p$?

As the title says, why is $$\wedge ^2 E[p] \cong \mu_p,$$ where $E[p]$ refers to the $p$-torsion points of an Elliptic curve over a number field $K$, $\wedge$ refers to the exterior product and $\...
Yang Awotwi's user avatar
1 vote
1 answer
36 views

Upper bound for modulus of the coefficients of a power series

On p.56 of the fourth edition of Serge Lang´s complex analysis book, he states that if a power series $\sum a_n z^n$ has radius of convergence $R>0$, then there exist a constant $C>0$ such that ...
Nerhú's user avatar
  • 301
2 votes
1 answer
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About the formal exponential power series over an arbitrary field of characteristic 0

Assume $K$is any field of characteristic $0$ which ensues that it makes sense to define in the ring $K[[[X]]$ of formal power series with coefficients in $K$ the formal exponential power series $exp(...
Udo Zerwas's user avatar
2 votes
0 answers
82 views

What is the most efficient way to find the coefficient before the $n$-th term in this generating function?

I am calculating the square of the Pascal matrix (i.e. the infinite lower triangular matrix with entries $A_{nk}={n\choose k}$) using the theory of Riordan arrays, and have obtained (or so I believe) ...
Daigaku no Baku's user avatar
1 vote
1 answer
90 views

if $f$ has a non-zero constant term, then it has an inverse as a power series

On p.40 of the fourth edition of Serge Lang´s complex analysis book, he states that if $f=\sum_{n\geq 0}a_n T^n$ is a formal power series with $\mathrm{ord}(f)=0$ (i.e. $f$ has constant term $a_0\neq ...
Nerhú's user avatar
  • 301
2 votes
0 answers
68 views

Convergence of infinite products of formal power series

Setting Let $R$ be a domain. The ring $R[[X]]$ of formal power series is a complete ultrametric space, see this Wikipedia article. According to the same source, "the philosophy of formal power ...
azimut's user avatar
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1 answer
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Cohomology of $BU(n)$

The cohomology ring for $BB\mathbb{Z}$ is $\mathbb{Z}[[x]]$, where $x$ lies in degree $2$. My question is about $\mathbb{Z}[[x_1, \dots, x_n]]$. I was trying to find a topological group (ideally an ...
user avatar
3 votes
1 answer
61 views

Statement unclear-"Cyclotomic fields and zeta values"

I am currently reading the book Cyclotomic fields and zeta values by J. Coates and Sujatha. There is a statement(Page 16) that is not clear immediately. Given $f$ in $R$, define $h(T)=\prod_{\zeta\in\...
user631874's user avatar
4 votes
1 answer
125 views

How to calculate Puiseux series for $x^{e^x}$?

Mathematica gives the result: $$x+x^2\ln(x)+\frac{1}{2}x^3(\ln(x)+\ln(x)^2)+\frac{1}{6}x^4(\ln(x)+3\ln(x)^2+\ln(x)^3)+...$$ I have searched for solved examples or step-by-step tutorials calculating ...
Mohamed Mostafa's user avatar
0 votes
0 answers
24 views

When a sequence of formal power series is summable. [duplicate]

I'm studying about formal power series in ring theory from Grillet's Abstract Algebra. I'll go first with some definitions. We don't have a topology here, so we can't use limits, and this is why these ...
Fabrizio G's user avatar
  • 2,117
3 votes
0 answers
168 views

Show that the ring of formal power series in a commutative ring $R$, $R[[x]]$ is noetherian.

Yes, I am aware that this has been answered (If $R$ is a Noetherian ring then $R[[x]]$ is also Noetherian), but the answers given did not answer my specific question regarding this: In my notes from a ...
Ben123's user avatar
  • 1,308
2 votes
0 answers
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Transcendence of meromorphic function vs formal power series

Consider the meromorphic function $f$ on $\mathscr D=\{z\in\mathbb C\mid|z|<1\}$ definied by $\displaystyle f(z)=\sum_{n\ge1}\frac{z^{2^n}}{z^{2^n}-\frac12}$. Obviously $f$ admits infinitely many ...
joaopa's user avatar
  • 1,157
1 vote
3 answers
103 views

Find a formula for summation of a increasing term (formal power series)

I am reading the exercises in the Combinatorics book written by Miklos Bona and trying to solve this exercise of 2.10. Find an formula for $a_n = \sum_{i=0}^{n-1} (i+1)a_i$ , where $a_0 = 1$ I've ...
lucifer Futonli's user avatar
5 votes
2 answers
190 views

Seeking more alternate proofs of a combinatorial generating function identity $G(x)=\overline{G}(-x)^{-1}$ related to counting strings.

Let $\mathcal{S}=[m]^*$ be the set of all strings on the alphabet $[m]=\{1, 2,\cdots, m\}$. Let $\Sigma\subset[m]^2$ be a set of strings of length $2$, and let $\overline{\Sigma}=[m]^2\backslash\Sigma$...
Christian E. Ramirez's user avatar
0 votes
0 answers
20 views

Definition of formal path in the group of diffeomorphisms

In M.Kontsevich's paper about deformation quantization https://arxiv.org/pdf/q-alg/9709040.pdf. Page3. He defines formal Poisson structure as the set of equivalence classes of Poisson structures ...
user24918's user avatar
3 votes
1 answer
223 views

Is $\mathbb{Z}_p \cong \mathbb{Z}[[X]]/(X-p)$?

Let $p$ be a prime number and denote by $\mathbb{Z}_p$ the ring of p-adic integers. Denote by $\mathbb{Z}[[X]]$ the ring of formal power series with integer coefficients and let $\frac{\mathbb{Z}[[X]]}...
greenbean's user avatar
1 vote
1 answer
105 views

Metric Induced by order of formal power series

In the book I'm following the order of a formal real power series $a=\sum_{n=1}^{\infty} a_nx^n$ is defined as: $$o(a)=\begin{cases} \infty, \, \, a=0\\ \min\{n:a_n \neq 0\}, \, \, a \neq 0 \end{cases}...
Tropax's user avatar
  • 373
0 votes
0 answers
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A certain formal power series equals the zeta function of a curve

Be $X$ a projective, geometrically connected and smooth curve over the finite field $\mathbb{F}_q$. Then one can define the following series, i.e. the zeta function of the curve: $$Z(t) = \exp\left(\...
Luca Morstabilini's user avatar
1 vote
0 answers
56 views

Isomorphism between quotients of two variable formal power series ring

$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$. Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$. So far, I've shown the following. $k[[x,y]]$ is an ...
isz's user avatar
  • 31
2 votes
1 answer
111 views

Freeness of the algebra of formal power series?

Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. Then the algebra of formal power series $A[[x]]$ can be viewed as a $\mathbb{k}[[x]]$-module in a natural way. My question is ...
Tison Cik's user avatar
3 votes
0 answers
110 views

Is the ring of formal power series a localization of some non local ring at prime ideals?

Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ ...
Eloon_Mask_P's user avatar
0 votes
0 answers
39 views

Evaluation of a rational function vs formal power series

Let $K$ be a complete valued field and $X$ be an indeterminate $K[X]$ be the ring of polynomials over $K$ in the indeterminate $X$ and $K(X)$ be its quotient field. For $a\in K$, consider the ...
joaopa's user avatar
  • 1,157
0 votes
0 answers
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notation for creating formal power series and $R-module$

The following is taken from "Module Theory an approach to linear algebra" by T.S Blyth $\color{Green}{Background:}$ $\textbf{(1) Example}$ Let $R$ be a unitary ring and let $R^N$ denote the ...
Seth's user avatar
  • 3,683
1 vote
1 answer
79 views

Affineness of the algebra of formal power series?

Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. It is clear that when $A$ is an affine $\mathbb{k}$-algebra (that is, $A$ is finitely generated as a $\mathbb{k}$-algebra), ...
Tison Cik's user avatar
0 votes
0 answers
32 views

Condition for convergence of series of formal power series

Proposition 1.1.8 in this answer, from "Enumerative Combinatorics" by Stanley, says The infinite series $\sum_j F_j(x)$ converges if and only if $\lim_{j \to \infty} \deg(F_j(x)) = \infty$. ...
mortimer's user avatar
2 votes
1 answer
188 views

Is a formal power series a fraction of polynomials?

We consider formal power series over a field. Power series such as $x+x^2+ x^3 +\cdots$ can be expressed as fraction of polynomials where the denominator is a unit in the power series ring. There are ...
Eloon_Mask_P's user avatar
0 votes
0 answers
17 views

Convergence of formal powers seires implies convergence of evaluation

For every $n\in\mathbb N$, let $f_n(Z)=\sum_{k\ge0}f^{(n)}_kZ^k\in\mathbb R[[Z]]$ be a formal power series. One assumes that there exists a positive real number $C$ such that for all $(n,k)\in\mathbb ...
joaopa's user avatar
  • 1,157
1 vote
0 answers
84 views

Lang's proof of Euclidean algorithm for power series

I have a question about the use of projections in Lang's proof of the Euclidean division algorithm for power series (Algebra - Serge Lang, Chapter IV, section 9, Theorem 9.1). Specifically, there is a ...
Zahra Abdullah's user avatar
0 votes
0 answers
73 views

Why do we care if a power series has roots

I am reading up on Christol's theoreom and an important part is that k-uniform transducers (where k is somehow related to prime numbers) preserve the algebricity of a formal power series (taking the ...
Mark Santolucito's user avatar
4 votes
0 answers
110 views

Krull Dimension of $\mathbb{C}[[X,Y]]/(Y^2)$

I want to compute the Krull Dimension of the ring $\mathbb{C}[[X,Y]]/(Y^2)$. I have tried the following and would be glad if somebody could verify or point out mistakes in the following proof: Let $y=...
Liva's user avatar
  • 61
3 votes
1 answer
90 views

Power series expansion of $\frac{kx}{(x+1)^k-1}$ at $x=0$

The question is simple: for integer $k\geq 2$, how to calculate the power series expansion of $f(x)=\frac{kx}{(x+1)^k-1}$ at $x=0$? As far as I am aware, there are two effective (operatable) ways to ...
Yijun Yuan's user avatar
0 votes
1 answer
38 views

Do pro-objects in a monoidal category have a completed tensor product?

Given a monoidal category C it seems natural to define a completed tensor product in its category of pro objects Pro(C) by $$("lim_{a\in A}" X_a) \hat\otimes ("lim_{b\in B}" X_b)="lim_{(a,b)\in A\...
dpistalo's user avatar

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