# 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### $\mathbb{K}[[x,y]]/\langle\, f\,\rangle \cong \mathbb{K}[[x]],\mathbb{K}[[y]]$. 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 $F_l(t)$ ...
### 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 ...