# 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.

418 questions
Filter by
Sorted by
Tagged with
30 views

### Lang's proof that $k[[X_1,...,X_n]]$ is UFD

I have trouble understanding the proof of Chapter IV Theorem 9.3 in Lang's Algebra, 3rd edition, where he proves that if $k$ is a field, then the ring of formal power series in $n$ variables is a UFD, ...
18 views

135 views

62 views

### Zero divisors of the formal power series ring $A[[x]]$

In class we started studying the formal power series ring $A[[x]]$ of a ring $A$ and I've been all day trying to find how the zero divisors of this formal power series ring $A[[x]]$ should be, if $A$ ...
98 views

### Is it possible to get positive radius of convergence for composition of two formal power series, if none of them has positive RoC. [closed]

I get counter example/proof for all other possibilities. But this one I couldn't do.
1 vote
33 views

### Quotient ring of formal power series

Let $R$ be a ring and $f \in R[[x]]$. Do we know anything about the ring $R[[x]]/(f)$. I understand that by making $R$ local and complete and adding some requirements on $f$, we can say a lot by using ...
36 views

35 views

### Formal Taylor Series Expansion

In this paper, the authors consider how to convert a system of $N$ ODEs to a PDE where $N \rightarrow \infty$ in some appropriate sense. In what follows $u_{n+1}$, the solution of the $(n+1)$th ODE, ...
1 vote
64 views

### Is $\frac{1}{x(1+x)}$ a well-defined generating function?

Is this a well-defined generating function? $$\frac{1}{x(1+x)}$$ We know that $\:\frac{1}{(1+x)} = \sum_{n \ge0}(-1)^nx^n$,$\:$ hence the notation $\frac{1}{x}\sum_{n \ge0}(-1)^nx^n \:$ would act as ...
1 vote
35 views

### Solving a system of nonlinear algebraic equations with power series

In a physics problem I am working on, I need to solve a system of nonlinear algebraic equations arising from truncating Taylor polynomials. I will outline the physical background for context, but my ...
74 views

### An infinite family of Artin-Schreier polynomials which all split in $\mathbf{F}_q(\!(\theta)\!)$

Let $\mathbf{F}_q$ be a finite field with $q$ elements and let $K$ denote the local function field $\mathbf{F}_q(\!(\theta)\!)$. Let $R$ be its valuation ring $\mathbf{F}_q[\![\theta]\!]$. Let $u$ be ...
57 views

### Taking the logarithm of a formal power series definition

I came across with the logarithm of a formal power series in a paper i am reading, but i have not found what is the definition of this 'operator(?)'. Can someone please let me know the definition and ...
1 vote
42 views

87 views

### Silverman AEC: Lemma IV.6.3 [on the convergence of formal logarithm]

In Silverman's Arithmetic of Elliptic Curves, Lemma IV.6.3(a) states that Let $R$ be a ring of characteristic $0$, complete with respect to a discrete valuation $v$, and let $p\in\mathbb{Z}$ be a ...
1 vote
80 views

### Silverman AEC: Corollary IV.4.4.

In Silverman's Arithmetic of Elliptic Curves, Corollary IV.4.4 states that (for an arbitrary ring $R$), Let $\mathcal{F}/R$ be a formal group and let $p\in \mathbb{Z}$ be a prime. There are power ...
42 views

1 vote
46 views

### Generating function of the succession $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$

I'm having serious trouble with this question. What is the generating function of $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$? I have thought first that I know some things related to this. Firstly, I know ...
45 views

### Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

I'm currently trying to understand the structure of quotients of power series rings, and found a particular example I'm confused about. Let $f = x-x^2$ be a polynomial in $\mathbb{Z}[[x]]$, and ...
85 views

### Order of a real formal power series

I am struggling to understand what the order of a real formal power series is. I am working through the book "Topology: An Introduction" by Stefan Waldmann. There he describes the "...
76 views

### Solving an implicit equation to find the exponential generating function of a specific sequence

Suppose we have a set $A$ with $n$ elements. First we partition $A$ in at least two blocks. Then, we partition each block in the previous partition that is not of size 1 in at least two blocks. We ...
127 views

### Generating function of a polynomial

Suppose I want to find a simple formula for the generating function of a general polynomial sequence $a_n=P(n)$. Obviously it is enough to find the generating function of the sequence $a_n=n^k$ for ...
1 vote
101 views

### Find the closed form for the generating fuction

I am given $g_0 = g_1 = \frac{1}{2}$ and $g_n + (n+1)g_{n+1} = \frac{1}{n!}$ where $n\geq2$. I need to find the closed form for the generating function $g(z)$ and the closed form for $g_n$. I'm not ...