# Questions tagged [power-series]

The tag has no usage guidance.

258 questions
Filter by
Sorted by
Tagged with
0answers
63 views

1answer
302 views

2answers
265 views

0answers
187 views

1answer
104 views

### Ideal in ring of power series

Let $K$ be a field of characteristic $p$ and $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring over $K$ such that $n, p \geq 3$. Consider the ideal $I$ defined by \begin{...
2answers
180 views

2answers
316 views

### Extracting Dirichlet series coefficients

Cauchy's integral formula is a powerful method to extract the $n$'th power series coefficient of an analytic function by evaluating a single complex integral. Is there any such analytic method to ...
0answers
35 views

### Algorithm to determine if a rational fraction has only non negative coefficients

Is there an algorithm that takes as input a polynomial in two variables $P \in \mathbb{N}[x,y]$ and outputs YES if and only if the coefficients of the series $\frac{1}{1-(x+y)} - \frac{1}{1-P}$ are ...
1answer
108 views

### Finite extension of $K[[X]]$ and the norm

Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$ $$f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.$$ ...
1answer
115 views

### From recursive polynomials to a $q$-series

Fix an integer $k$. Let $b_n(q)$ be the polynomial defined by the recursive equation $$b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$$ initializing with $b_0(q)=0$. I run into the ...
0answers
82 views

### Simple transfinite generalization of $p$-adic integers

One way to define the ring of $p$-adic integers is as a quotient of the formal power series semiring $\Bbb N[[x]]/(x-p)$. One can likewise start with the formal power series ring $\Bbb Z[[x]]/(x-p)$ ...
0answers
20 views

1answer
174 views

### Are the ring of power series and the ring of germs of holomorphic functions catenary?

I am wondering if the following rings are catenary: If $k$ is a field, is the ring of formal power series $k[[X_1,\dots,X_n]]$ catenary? Is the ring of complex power series with a non-zero radius of ...
1answer
204 views

0answers
122 views

### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity. If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
0answers
101 views

### On a certain $(-1)$-Eulerian polynomials of type $B$

Let $(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q)_0:=1$. Define a $q$-exponential by $$e_q(z)=\sum_{n\geq0}\frac{z^n}{(q)_n}.$$ There is a notion of $q$-Eulerian polynomials of type $A$, see the ...
1answer
215 views

0answers
220 views

2answers
622 views

0answers
89 views

### Relation between coefficients of expansions

Related to Relations between coefficients of expansions of a rational function at 0 and infinity I commented at the linked question that the question seemed less about what happened "at infinity", ...
1answer
470 views

### Relations between coefficients of expansions of a rational function at 0 and infinity

This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up." Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and \$B(t)=\sum_{k>M}...