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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9 views

Sum with reciprocal general term.

Given a function expressed as a formal power series as follow $$ f(x)=\sum_{k\geq0}a_kx^k, $$ I was wondering if there exist some way to relate the latter with a function $g$ $$ g(x)=\sum_{k\geq0}\...
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23 views

Does $\ln$ belong to the field of Hahn series or not?

Consider the field of Hahn series (see e.g. https://en.wikipedia.org/wiki/Hahn_series )$$K[[T^{\mathbb{Q}}]]=\left\{\sum_{q\in\Bbb{Q}} a_q T^q: a_q\in K,\text{supp}(a_q)\text{ is well ordered} \right\}...
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48 views

“Type” of a permutation, set function, partition

Suppose I take the permutations on $n$ letters, $\mathcal{S}_n$, and consider two functions from it $$f:\mathcal{S}_n \xrightarrow{\substack{\text{count} \\ \text{number} \\ \text{of cycles}}} \{1, \...
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22 views

How to prove an exercise from Frank Olver's book

I'm reading Frank Olver's book, called Asymptotics and Special Functions. There is an difficult exercise. Problem. Suppose that $f,\frac{1}{f}$ possess the following asymptotic expansions : $$f(z)...
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18 views

When a power series $f(x) \in x O_K[[x]]$ with $f'(0) \not\equiv 0 \mod m_K$?

Let $K$ be the finite extension of $p$-adic field $\mathbb{Q}_p$ with algebraic closure $\bar K$. Let $O_K$ be the ring of integer and let $m_K$ be the maximal ideal. Consider a power series $f(x) ...
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1answer
42 views

The criteria of commuting two formal power series,

Let $f = x+\sum_{n=2}^{\infty} a_n x^n$ and $g = x+\sum_{k=2}^{\infty} b_n x^n$ be two formal series without constant term. Then $$ f \circ g(x) =x+ \sum_{k=2}^{\infty} b_k x^k+\sum_{n=2}^{\infty} a_n ...
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0answers
17 views

Simplifying a generating function with $\exp$

In Stanley's Enumerative Combinatorics vol 2, during the solution to exercise 7.69, it says: $$\exp \sum_{n\ge 1} \frac{1}{n} \bigg(\sum_ix_i^n+\sum_{i<j}(x_ix_j)^n \bigg)=\frac{1}{\prod_i (1-x_i) ...
2
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1answer
24 views

Formal power series: are subgroups of the multiplicative group closed?

I have a subgroup $G$ of the multiplicative subgroup of the ring of formal power series on n coefficients, $(\mathbb{Z}[[X_1,...,X_n]])^*$, and an element $a = \sum_{i \in I} a_i X_1^{i_1} ... X_n^{...
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1answer
35 views

Formal power series are a euclidean ring

Denote by $F[[T]]$ the ring of formal power series over a field $F$ (i.e expressions of the form $\sum_{n=0}^{\infty}a_nT^n$, $a_i \in F$). I need to show that this is a euclidean ring with respect ...
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2answers
45 views

Power Series of a sin of a power series

I was wondering if there exists a power series for the sin of a power series, in other words: which is the formula for the coefficients $\xi_{\lambda}$ in terms of the $f_{\lambda}$ in the expansion: ...
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19 views

How to calculate $f^{-1}$ of a formal power series $f \in R[[x]]$ ??

Let $R[[x]]$ be the ring of formal power series over the ring $R$. Let $f(x), g(x)\in R[[x]]$ be such that $f(x)=x+ax^m $ mod deg($m+1$) and $g(x)=bx+cx^m$ mod deg $(m+1$) and assume $f(x)$ be ...
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1answer
30 views

Show there exists a unique formal power series $g$ such that $g^d=f$

The problem is as follows: 'Let $f$ be a formal power series with $f_0=1$. Show that there is a unique formal power series $g$ with $g_0=1$ such that $g^d=f$, for some positive integer d.' I am sure ...
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1answer
18 views

How to prove that a power series in a unital Banach algebra converges normally in an open ball around 0

Let $A$ be an unital Banach algebra over a vector space $X$. Let us consider the power series: \begin{equation*} \sum_{k=0}^{+\infty} c_k x^k \end{equation*} with coefficients in $\mathbb{K}$ and ...
4
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1answer
76 views

Is $R[[x]][[y]]$ the same as $R[[x,y]]?$

Let $x,y$ be two central indeterminates in $R$, is $R[[x]][[y]]=R[[x,y]]$? My take on this is: Let $f(x,y) \in R[[x]][[y]]$, then $f(x,y) = \sum_{i=0}^{\infty}g_i(x)y^i = \sum_{i=0}^{\infty}\sum_{j=0}...
1
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1answer
26 views

Help Understanding Proof Involving Formal Power Series

Let $R$ be a ring, and let $f, g \in R[[t]]$. As part of a proof that I am reading, we have the following setup: $$ \begin{align} f(0) &= 1\\ g(0) &= 0\\ p f(t) &= f(g(t))\cdot g'(t) \...
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0answers
34 views

Ideal of formal power series over $\mathbb{R}$

I found this solution Ideals of formal power series ring. I can follow the proof, but I'm a bit confused on what $a$ is. It seems as if in the solution, to define $a$, we need to know what $I$ is ...
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0answers
76 views

Infinite product of polynomials what does it mean?

What is the canonical way to understand such things as $$(1+q + q^2 + \ldots q^9)(1+q^{10} + q^{20} + \ldots q^{90})((1+q^{100} + q^{200} + \ldots q^{900}) \ldots \quad ?$$ It is a problem from a ...
0
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2answers
33 views

Existence of ring homomorphism from $\mathbb{Z}[[x]] \to \mathbb{Z} $ [closed]

Let $\mathbb{Z}[[x]]$ denote the ring of formal power series with coefficients in $\mathbb{Z}$. Can there exist a ring homomorphism $$\Phi : \mathbb{Z}[[x]] \to \mathbb{Z}$$ such that $\Phi $ sends $...
5
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1answer
95 views

$k$-algebra morphisms from formal power series ring

Background/Motivation: I was playing around with a certain construction that I am trying to generalize and therefore needed to compute some examples to get a feel for the situation. I realized that I ...
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2answers
33 views

Given a formal power series $f_1$, can we produce another formal power series $f_2$ suing computer program s.t. $f_1 \circ f_2=f_2 \circ f_1$?

When two formal power series commutes ? Algorithm: Let $f = \sum a_n x^n$ and $g = \sum b_n x^n$ be two formal series without constant term. Then $$ f \circ g = \sum a_n g(x)^n = \sum a_n \left(\sum ...
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0answers
111 views

Unusual completion of $\Bbb{Z}_p[x]$

The $p$-adic norm $$|\sum_{n=0}^N a_n x^n|= \sup_{n\le N} |a_n|_p$$ is an absolute value on $\Bbb{Z}_p[x]$. The completion, call it $S_1$, is the ring of formal power series whose coefficients $\to 0$,...
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1answer
66 views

Questions about evaluation map from $\mathbb Z[[X]]$ to $\mathbb Q$ and $\mathbb Z[[X]]$ to $\mathbb C$.

Let $ℤ[[𝑋]]$ denote the ring of formal power series with coefficients in $ℤ$. Question 1: If $\alpha \in ℚ$ is unit such that $\alpha \notin ℤ$, then can there exist a ring homomorphism $\phi :ℤ[[𝑋...
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1answer
62 views

Existence of Ring homomorphism from Formal Power series ring to a ring.

Let $R$ be a ring, and $S$ be a subring of $R$. Denote $S[[x]]$ for a ring of formal power series with coefficients in $S$. Let $\alpha \in R$ be a unit, such that $\alpha \notin S$. Can there exist a ...
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0answers
19 views

Understanding the solution to ODE using differential operator theory with basic algebra

Consider the following constant-coefficient ODE with $f\in C^\omega(A)$, where $A$ is an interval: $$a_n\frac{d^n}{dx^n}y+\cdots+a_0y=f(x)$$ It can be written as $$P(D)(y)=f(x),$$ where $P$ is a ...
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0answers
107 views

Computing Hilbert-Samuel multiplicity of $k[[f_1(t),…,f_n(t)]]$

Let $k$ be a field and $k[[t]]$ be the formal power series ring over $k$. Consider the subring $R=k[[f_1(t),...,f_n(t)]]$ for some $f_1(t),...,f_n(t)\in k[[t]]$ such that $k[[t]]\subseteq Q(R)$ and ...
1
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1answer
51 views

On embedding $\mathbb C[x_1,…,x_d]/P $ inside $\mathbb C[[T]]$

Let $P$ be a prime ideal of $\mathbb C[x_1,...,x_d]$ such that ht$(P)=d-1$ i.e. $\dim (\mathbb C[x_1,...,x_d]/P)=1$. Then is it necessarily true that there exists an injective $\mathbb C$-algebra ...
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0answers
22 views

Brent and Kung's algorithm for fast composition of formal power series.

I'm reading page 498-499 of textbook and 2.2 of original paper to understand the Brent and Kung's algorithm for fast composition of formal power series. We consider the formal powerseries at $\bmod z^...
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0answers
48 views

Is $\sum_{i=0}^n a^{n-i}b^ix^i$ a regular element of $R[[x]]$ when $ax + b$ is regular?

$R$ is a ring and $R[[x]]$ is the ring of formal power series over $R$. I'm trying to get a better understanding of how zero divisors in $R[[x]]$ behave. Here's a simple question to this effect. ...
0
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0answers
29 views

Convergence of Newton's method for inverse function

I follow page 8 of this slide. I want to know the convergence of Newton's method of inverse function of formal power series. In order to get the inverse function of $f$,$f^{-1}(X)$, we want to ...
4
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1answer
115 views

Formal Power Series as Initial Objects?

We often apply formal power series in places where it seems, at face value, somewhat suspect to do so. I'm primarily interested in why these formal manipulations work so broadly. A prime example ...
2
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1answer
110 views

Formal group laws VS Formal Lie groups

Warning: this is a soft question about usual terminology, to make sure I understand things correctly. Let $R$ be any commutative ring and $n\geq 1$. Consider the $R$-algebra $\mathcal A = R[[X_1,\...
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1answer
83 views

Integrality of coefficients in some formal power series

Suppose we have a formal power series $g(x)$ with integer coefficients such that $g(0)=1$. Suppose also that there exists an invertible formal power series $f(x)=x+(\text{higher order terms})$ such ...
3
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2answers
46 views

Represent $f(x)$ with $g(x)$ when the taylor expension has specific dependency

If I have $f(x)$ and $g(x)$ like that:$$f(x)=\sum_{n=1}^{\infty} a_nx^n$$$$g(x)=\sum_{n=1}^{\infty} \frac {a_nx^n} {n}$$How can I find u(x) such: $f(u(x))=g(x)$?I also know that the series $(a_n)$ is ...
3
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1answer
100 views

Multivariate Lagrange inversion with zero powers

(also asked in MO) The multivariate Lagrange inversion formula, which I found in a couple of papers (such as this and this), is as follows. If $f_i=t_ig_i(f)$, $1\le i\le k$, then $$[\boldsymbol{t^n}]...
4
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1answer
83 views

Dimension of the power series ring localized at its variables

I am trying to answer the question Let $k[[X_1,\ldots,X_n]]$ be the power series ring in $n$ variables over a field $k$. What is the (Krull) dimension of $k[[X_1,\ldots,X_n]][(X_1\cdots X_n)^{-1}]$,...
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0answers
22 views

Form of an analytic function

Let $K$ be a field of characteristic zero complete with respect to a non archimedian absolute value with a residue field of characteristic $p>0$. Let $\mathcal{H}^\dagger=\cup_{\varepsilon >0} \...
3
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1answer
63 views

Find the closed form for the following power series

I wish to find the closed form for this power series: $1+\dfrac{1}{6}x+\dfrac{3}{40}x^2+\dfrac{5}{112}x^3+\dfrac{35}{1152}x^4...$ I have been able to spot the that the second term is the first term ...
2
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0answers
46 views

Finding a generating function for a formal power series containing $n$ and $a_n$

Given the sequence $(a_n)_{n\geq0}$, the elements of which are recursively defined as follows: $$a_0 := 0; a_{n+1} := n \cdot a_n + 1$$ We can define a formal power series $A(z) = \sum_{n\geq0}a_nz^...
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1answer
33 views

Can this be a Schauder basis of $\mathbb{R}[[x]]$

A few hours ago, I asked a question about using Taylor expansion of two analytic functions on $\mathbb{R}$ to determine whether these two functions are linearly independent. Basically I was trying to ...
0
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1answer
46 views

Power Series and Differentiated Power Series

What I want to show: Let $\sum^{\infty}_{n=0} a_nx^n$ be a power series. Then $\sum^{\infty}_{n=0}na_nx^{n-1}$ has the same radius of convergence. Proof: Suppose $\sum^{\infty}_{n=0} a_nx^n$ ...
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0answers
32 views

A question relating to certain algebraic manipulation of a formal power series written in the form of infinite product

Suppose there is formal power series in infinite product form as follows: $$\prod_{d\geq 1} \left(1+\frac{u^d}{q^d-1}\right)^{a_d}$$, where $a_d$ are positive integers. Consider the expression $$\...
2
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0answers
56 views

Finitely generated module over formal power series can be transformed in a free module.

If $M$ is a finitely generated $\mathbb{C}[[ t]]$-module, I have to show that there exists a $n \in \mathbb{N}$ such that $t^nM$ is a free $\mathbb{C}[[t]]$-module. I have no idea how I can start with ...
0
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1answer
32 views

$\mathbb{K}[[x,y]]/\langle\, f\,\rangle \cong \mathbb{K}[[x]],\mathbb{K}[[y]]$. in formal power series ring

Let $\mathbb{K}[[x,y]]$ be the ring of formal power series in $x,y$. Now let $f(x,y) = \sum_{i+j \geq 1} a_{i,j}x^iy^j \in \mathbb{K}[[x,y]]$ so that $a_{1,0}$ or $a_{0,1}$ is not zero. I would like ...
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0answers
41 views

Is there a power series generating function for Euler's totient function

is there a closed form for the expression $\sum_{n \geq 1} \phi(n) x^n$ ? Everybody knows about the Dirichlet generating function for $\phi$ but I can't find anywhere the power series generating ...
0
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1answer
61 views

Ring of formal power series over $\mathbb{R}.$

I am studying ring of formal power series over $\mathbb{R}$ and trying to solve this following exercise. The ring of formal power series over $\mathbb{R}$, denoted $\mathbb{R} [[ x ]]$, is a ring ...
0
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0answers
23 views

What is the definition of stable power series?

What is the definition of stable power series? I have found it in a paper which says: A power series $h(X) \in X \cdot K[[X]]$ is said to be stable if neither $h'(0) \neq 0$ nor $h'(0)$ is a root ...
1
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2answers
45 views

Prove that $(1 + z)^x \cdot (1+z)^y = (1+z)^{(x+y)}$ for formal power series

Imagine that I define a function by formal power series: $$ (1 + z)^x := \sum_{n = 0}^{\infty} {x \choose n} z^n $$ where ${x \choose n} := \frac{x \cdot (x - 1) \ldots \cdot (x - n + 1)}{n!}$ How ...
3
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0answers
36 views

Generalizing the Weierstrass Preparation Theorem to formal power series in multiple variables

The statement of the Weierstrass Preparation Theorem is as follows: Let $f = \sum_{i=0}^\infty a_iX^i \in K[[X]]$ for some field K where $a_h \neq 0$ and for every $n < h$, $a_n = 0$. Then the ...
0
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0answers
34 views

Can we consider formal power series $P(t)=\sum_{m=0}^{\infty} \binom{2m}{m} \left[m^l(4t-1)^l+F_l(t) \right]t^m$ as formal group law?

Can we consider the formal power series $P(t)=\sum_{m=0}^{\infty} \binom{2m}{m} \left[m^l(4t-1)^l+F_l(t) \right]t^m$ as a formal group law in 1 variable ? Where $l \geq 0$ are integers and $F_l(t)$ ...
0
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0answers
20 views

Explain the arrow or relation $ \text{Lie groups} \to \text{formal group laws} \to \text{Lie algebras}.$

Can you please explain why $\text{formal group laws}$ are intermediate (middle man) bewteen $\text{Lie groups}$ and $\text{Lie algebras}$? i.e, explain $$ \text{Lie groups} \to \text{formal group ...

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