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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Quotient of formal power series ring

Let $k$ be an algebraically closed field of characteristic zero, and let us consider the formal power series ring $k[[x]]$. What is the quotient $k[[x]]/x^{n}$ ?
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126 views

What is the universal property of the algebra of formal power series over a commutative ring?

Let $A$ be a commutative ring. For every set $I$, $A^{\mathbb{N}^{(I)}}$ is the algebra of formal power series. Suppose $\sigma:I\rightarrow J$ is a bijection. Ignoring topology, what is the canonical ...
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34 views

Extended ideal in formal power series ring is always equal to kernel of canonical reduction homomorphism $R[[X]] \rightarrow (R/I)[[X]]$?

Let $R$ be a commutative ring with unit element, let $I$ be an ideal in $R$. Let $A = R[[X]]$, the ring of formal power series with coefficients in $R$. Let $I_A$ be the ideal generated by elements of ...
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1answer
31 views

What is the formal power series in Shermann-Morrison formula?

I'm learning about the Shermann-Morrison formula and the way to find the inverse matrix uses a formal power series. My question is how is this formal power series done? I read in some websites and I ...
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22 views

Can you use the inverse Z transform to compute a power series?

I know one way to compute a power series is to use a Taylor Series and compute derivatives. I'm wondering if you can also do it using the Inverse Z Transform? It looks like yes, but I'm getting stuck ...
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24 views

Generating Function where $x^n$ is replaced with $x^n/(1-x^n)$

Suppose $\phi(x) = \sum_{n=1}^\infty a_n x^n$. If we know the closed form expression for $\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n}$ is there a way to find the closed form expression for $\phi(x)$?
2
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1answer
72 views

Power Series where partial sums are irreducible polynomials?

I want to show that there exists some formal power series, $f(x)\in\mathbb{Z}[[x]]$, such that each consecutive partial sum is irreducible in $\mathbb{Z}[x]$. Rewording this in terms of polynomials, I ...
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22 views

structure of ideals in formal power series

Let be $R$ a commutative unitary ring. If $R$ is a field then the ideals of $R[[t]]$ are only the ideals generated by $t^{n}$. If $R$ is not a field, but for example a $\mathbb{C}$ algebra?
2
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1answer
72 views

Division in $A[[x]]$ [duplicate]

I was looking for a division in a ring of formal power series. Specifically, let be $A$ a commutative ring with unit. Take $A[[x]]$ and $f\in A[x]$ a monic polynomial not invertible in $A[[x]]$ is so ...
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1answer
62 views

Is $R\otimes\mathbb{K}[[t]]$ isomorphic to $R[[t]]$?

Let $K$ be a field of characteristic zero. It's true that if $R$ is a $K$-algebra then $R\otimes_{\mathbb{K}}\mathbb{K}[[t]]\cong R[[t]]$ with the natural inclusion inducted by $x\otimes t^j$$\mapsto$$...
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1answer
32 views

Torsion-free Cohen-Macaulay modules

We say an $R$-module $M$ over integral domain $R$ is a torsion-free module if zero is the only element annihilated by some non-zero element of the ring $R$. Let $R=K[[x_1,\dots ,x_d]]$, $d>1$, be ...
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37 views

limiting process in a sequence of formal power series

I am reading https://core.ac.uk/download/pdf/82317236.pdf An introduction to algebraic deformation theory by Thomas F. Fox. On page 23, Theorem 3.1, the author has used some limiting process for ...
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1answer
125 views

Infinite product of infinite sums of formal power series: proof?

Teaching a course on algebraic combinatorics has made me aware of a technical fact about formal power series that is used throughout the subject, but that I have never seen formally stated, let alone ...
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1answer
73 views

Torsion-free modules over formal power series rings

Let $R=K[[x,y,z]]$ be the formal power series ring over a field $K$. We set $M=K[[x,y]]$. Then $M$ has a structure of $R$-module by $f:R\longrightarrow M $ via $g(x,y,z)\rightsquigarrow g(x,y,0)$. In ...
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2answers
101 views

Is there a closed form for $\sum \frac{1}{k!}\binom{m}{k}\binom{n}{k}x^k$?

Let $F = \sum_{k=0}^\infty \frac{1}{k!}a_kx^k$ be a formal power series and define $F_n = \sum_{k=0}^\infty \frac{1}{k!}\binom{n}{k}a_kx^k$. Is it possible to define $F_n$ starting from $F$ by some ...
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1answer
60 views

A basis for formal laurent series

I was looking for a countable $A$-shauder's basis for the Laurent formal series in two variables $ \mathbb{C}[[t,s]][(ts)^{-1}]$. $A=\mathbb{C}[[t-s]]$. For example $\{t^ns^n, t^{n+1}s^n\}_{n \in \...
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46 views

Is $X+Y+XY$ and $X+Y-XY$ is isomorphic formal group law over integer ring?

I would like to check whether $X+Y+XY$ and $X+Y-XY$ is isomorphic formal group law over integer ring $\Bbb{Z}$ or not. It is known that it is isomorphic over rational field. But what about in $\Bbb{Z}$...
3
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1answer
150 views

How to confirm $\phi(F_1(x,y))=F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$.

This question is from Silverman's 'the arithmetic of elliptic curves',$p134$. Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \to E_2$...
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40 views

Exercise about tensor product

Let be $R$ a $\mathbb{C}-$ algebra and take $R\otimes_{\mathbb{C}}\mathbb{C}[[t]]$, Is this isomorphic as ring to $R[[t]]?$ I think that I can prove it using the quozient of these ring and take the ...
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166 views

Functional equations satisfied by both sine and tangent functions.

The functional equation identity, assuming also $\,f(-x)=-f(x),\,$ $$ f(a)f(b)f(a\!-\!b)+f(b)f(c)f(b\!-\!c)+f(c)f(a)f(c\!-\!a)+ f(a\!-\!b)f(b\!-\!c)f(c\!-\!a) = 0 $$ has solutions $f(x)=k_1\sin(k_2\...
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1answer
36 views

Showing two completions are the same.

The problem is from Matsumura 8.5. Let $A$ be a Noetherian ring and $I$ a proper ideal of $A$. Consider the multiplicative set $S=\{1+a: a \in I \}$. Then $A_{S}$ is a Zariski ring with ideal of ...
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25 views

Why formal power series is in a object of the category of formal scheme?

What is an object and an morphism of the category of formal scheme? I heard formal group is an group object in a category of formal scheme. I wonder why formal power series is in a object of formal ...
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1answer
46 views

What is the difference between formal group and formal group law?

I heard a formal group law is a formal group with chosen coordinate. But I cannot grasp this meaning. My understainding; (1-dimmensional)formal group and formal group law are both element of formal ...
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37 views

Formal power series on polynomial ring and polynomial on formal series

I was reading about the formal power series in several variables and now I have some question. First question: Let be t and s two variables \begin{equation} \mathbb{C}[[t]][s]=\mathbb{C}[s][[t]]? \end{...
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1answer
46 views

Which Schauder bases of $F[[x]]$ have the multiplication property?

Both $\{\frac{x^n}{n!}:n\in\mathbb{N}\}$ and $\{\frac{x^n}{n^2+1}:n\in\mathbb{N}\}$ are Schauder bases for the ring of formal power series $\mathbb{R}[[x]]$ as a topological vector space over $(\...
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0answers
52 views

Inverse of formal multiplicative group

I am reading about the formal multiplicative group, with addition given by $F(x,y)=x+y+xy$, and I am wondering if there is a nice way to describe the inverse of an element. So if I let $x+y+xy=0$, ...
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1answer
38 views

Is the direct sum decomposition of formal power series provable without choice?

If we view the ring of formal power series $F[[x]]$ as a vector space over $F$, and we view the polynomial ring $F[x]$ as a subspace of $F[[x]]$, then the axioms of choice implies that $$F[[x]]=F[x]\...
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1answer
81 views

Do falling factorials form a Schauder basis for formal power series in some topology?

We usually talk about $F[[x]]$, the set of formal power series with coefficients in $F$, as a topological ring. But we can also view it as a topological vector space over $F$ where $F$ is endowed ...
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1answer
106 views

Proof of $C_n = \frac{1}{n + 1} \binom{2n}{n}$

I want to comprehend the first proof of $C_n = \frac{1}{n + 1}\binom{2n}{n}$ on Wikipedia. Link: here. I have problems with two steps: $$\sum_{n=0}^{\infty} \binom{\frac{1}{2}}{n}y^n = \sum_{n=0}^{\...
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31 views

torsion element of group associated to formal group

Let $R$ be complete local ring $M$ be the maximal ideal of $R$ $F$ be a formal group defined over $R$, with group law $F(X,Y)$. According Silverman's book 'the arithmetic of elliptic curves', example ...
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79 views

Is formal group a pair wise concept?

In Silverman's 'the arithmetic of elliptic curves', Formal group is defined as a power series which satisfies some conditions. But the book also reads the formal group like a pair, $(\mathrm{F},F)$. ...
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0answers
31 views

What topologies on the ring of formal power series make it a topological ring?

The standard topology on $\mathbb{R}[[X]]$, the ring of formal power series, is the $I$-adic topology, or equivalently the product topology on $(\mathbb{R},discrete)^\mathbb{N}$. This makes $\mathbb{...
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0answers
57 views

Group associated to multiplicative group

Let formal multiplocative group be $$\mathbb{G}_m(X,Y) = X + Y + XY$$ Let $R$ be a complete local ring, and $M$ be its maximal ideal. According to Silverman's 'the arithmetic of elliptic curves', ...
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1answer
28 views

Generating functions, the product between two formal power series with different denominators

I'm a bit lost in trying to take the product of a power series between two functions $f(x) = \frac{1}{(1-x)^k}$ and $g(x)=\frac{1}{(1-x^r)^k}$. I know both can be expanded to, \begin{align*} \frac{1}{(...
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0answers
21 views

Existence of a closed-form for “compression” of formal power series

Let "$k$-compression" ($k\in\mathbb Z, k\geq 2$) of formal power series be a map $\mathcal C_k : R[[X]] \to R[[X]]$ such that $\sum_{n=0}^{\infty} a_n X^n \mapsto \sum_{n=0}^{\infty} a_{n \...
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0answers
38 views

What is the relation between formal group and formal scheme

What is the relation between formal group and formal scheme? Formal group is power series, which behaves like ' a group law without any group elements'. Is former one is special case of latter one?
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33 views

Sum of convergent power series with increasing degrees is convergent.

I'm contemplating Exercise 1.1.3 from Greuel, Lossen and Shustin's Introduction to Singularities and Deformations. It states Let $(f_k)_{k \in \mathbb N}$ be a sequence of convergent power series ...
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37 views

Additive power series in positive characteristic

Let $K$ be a field of positive characteristic $p$. Consider a power series $F\in K[[X_1,\dots,X_n]]$ that is additive, that is, $$F(X_1+Y_1,\dots,X_n+Y_n)=F(X_1,\dots,X_n)+F(Y_1,\dots,Y_n)\in K[[X_1,\...
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2answers
47 views

Determining all formal power series satisfying the equation $Q^m=(1+x)^n$ for $n,m\in\mathbb{Z}\setminus\{0\}$

The question: Let $m$ and $n$ be integers $\neq 0$. Determine all solutions $Q=1+b_1x+\cdots$ of the equation $$Q^m=(1+x)^n.$$ My current attempt at a solution: Let $Q\in\mathbb{F}[[x]]$. The formal ...
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0answers
40 views

Factorization of complex formal power series into linear factors

It is well-known that, for every $ n \in \mathbb{N}^+ $ and $ c \in \mathbb{C}^{n + 1} $ such that $ c ( 0 ) \neq 0 $, there exists $ r \in \mathbb{C}^n $ such that $ \displaystyle\sum_{k = 0}^n \left(...
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0answers
24 views

Generating function of a sequence without leftshift

Since $x$ is not invertible in the ring $R[[x]]$ of formal power series over $R$ how can I find generating function of: $a_n = 0,2,0,4,0,8,0,16,0,...$ without using formal Laurent series ( $R((x))$ )? ...
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57 views

Intuitive explanation why “shadow operator” $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } \frac{e^{-iwx}}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^{i ...
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1answer
63 views

An analytic function on $\Bbb{C}$ has infinite radius of convergence about any point

I may be overlooking a small detail, but this question has me stumped. Suppose $f:\Bbb{C}\to \Bbb{C}$ is analytic. Then, by definition of analyticity, for every $z_0\in \Bbb{C}$, there exists a formal ...
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50 views

A question is about calculation on the coefficients of power series by recurrence relation

The following question is about little calculation on the coefficients of power series and recurrence relations: Let $R$ be a commutative ring with a subring $S$ and $I$ be an ideal of $S$. Let $p \...
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1answer
50 views

Completion of a surface along a curve.

Let $R$ be the ring $\mathbb{Z}[x,y,z]/(x(y^2-4z)-y^2+3z)$. I am trying to prove that the completion of this ring at the ideal $(x)$ is given by $\mathbb{Z}[[x]][y,z]/(x(y^2-4z)-y^2+3z)$. I am not ...
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2answers
93 views

Formal power series and the quadratic formula

I was recently reviewing a derivation of getting a formula for Catalan numbers in closed form. I understand the mechanics of the derivation, with the exception of one significant step: If $f(x) = c_0 +...
3
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1answer
85 views

Problem with a Arithmetico-Geometric Series

Good afternoon to everyone, I have the following question: What does the arithmetico-geometric series: $$S = \sum^{\infty}_{n=1} ne^{-nrt}$$ Converge to? ($r > 1$, $t > 1$) I tried to break it ...
2
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1answer
58 views

Can all linear operators be represented as functions?

Can we correspond any linear operator $L$ to a function $\phi(x)$ such that $$L f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\phi(\omega) \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, ...
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1answer
49 views

A formula for operator $\frac D{e^D-1}$?

Is there a good unambiguous formula for the linear operator $\frac D{e^D-1}$? I mean, $$x^a\to B_a(x)$$ $$\ln x \to \psi(x)$$ $$e^x\to\frac{e^x}{e-1}$$ etc.
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2answers
63 views

Where do the powers go?

I have this weird identity between power series. It's kind of like the relation between a geometric series and $\frac{1}{1-x}$. I was wondering if there was some theory developed along these lines I ...

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