Questions tagged [power-series]

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0answers
63 views

Relations between the Fourier coefficients and the Taylor coefficients of the function $e^{\left(\frac{1+z}{1-z}\right)^2}$

Let $\displaystyle f(z)=e^{\left(\frac{1+z}{1-z}\right)^2}$, $z\in \overline{\mathbb D}\setminus\{1\}$. Then the Taylor series $ f(z)=\sum_{n=0}^\infty a_nz^n$ for $f$ diverges everywhere on $\...
0
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0answers
60 views

Bound of Coefficients of Fourier Series of Composition

Let $f(x) = \sum_{n=0}^\infty f_ne^{inx} + \bar{f_n}e^{-inx}$ and $g(x) = \sum_{n=0}^\infty g_ne^{inx} + \bar{g_n}e^{-inx}$ where $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$. Both ...
-1
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0answers
88 views

Finding the number of real roots of a power series

Suppose we have a power series $f(x) = \sum_{i=0}^\infty a_nx^n$ that converges for all reals where $f(\pm1) \neq 0$. We want to find the number of real roots. In order to do that we find the roots ...
3
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0answers
65 views

Surjective maps between power series rings

Suppose that $A$ is a complete neotherian local ring and that we're given a surjective homomorphism $f: A[[x_{1}, \ldots, x_{n}]] \rightarrow A[[t_{1}, \ldots, t_{m}]]$. Can we always find a ...
3
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1answer
176 views

Power series equation with solution $1/e$ [closed]

As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root. Are there classical equations of the form $$\sum_{i=0}^{\infty} a_ix^i =1$$ that have $e$ ...
3
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0answers
31 views

Matrix series with Hadamard products

Let $A$ and $B$ be hermitian matrices (a special case that would already help would be $A^{-1} = B^T$). I'm looking for a closed form of the series $$X := \sum_{n=0}^\infty A^n \circ B^n$$ where $\...
7
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1answer
302 views

A ring of generalized power series

Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like $$ \sum_{n=0}^{\infty} a_n x^{b_n} $$ where $a_n\in \...
6
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0answers
78 views

Coefficients of some infinite product power series

Let $f(n)\colon \mathbb{P}\to\mathbb{R}_{>0}$, where $\mathbb{P}=\{1,2,\dots\}$, be some ''nice'' function such that $f(n) \to \infty$ as $n\to\infty$. For instance, $f(n)=1+\log(n)$ or $f(n)=n$. ...
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0answers
163 views

Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
10
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1answer
226 views

Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$. Indeed let $P(X)=\...
6
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2answers
265 views

Existence of radial limits of products of certain power series and $1-x$

Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
5
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1answer
220 views

Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients

Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
1
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0answers
187 views

Is there function that can be expanded as infinite power series with bounded positive coefficients?

Is there a rational function $F$ which may be expanded as power series with coefficients of unperiodical positive integers in such a form: $$F(x)=\sum_0^{\infty}a_i x^i,\qquad a_i\in \mathcal{N} \cup ...
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0answers
37 views

Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
8
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0answers
115 views

Padé Approximants of Power Series with Natural Boundaries

Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
1
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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{...
0
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2answers
180 views

Power series ring and monomials

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables. For a given positive number $\epsilon > 0$ we call a monomial $X_{...
9
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2answers
460 views

Involutions in $\mathbb{F}_p[[x]]$

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$? Here involution in $A[[x]]$ means $f\in ...
5
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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 ...
0
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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 ...
0
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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$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...
2
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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 ...
3
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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 $\Bbb Z[[x]]/(x-p)$ ...
0
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0answers
20 views

Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation $$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$ as $y\...
6
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1answer
145 views

Sufficient conditions for the coefficients of a generating function to dominate those of its square

Let $f(z)$ be a generating function (so in particular, its power series coefficients are nonnegative). I am interested in conditions which would ensure that for every $n$, the coefficient of $z^n$ in $...
3
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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 ...
2
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1answer
204 views

Étale fibration for $K[[X_1,…,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
3
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0answers
78 views

Measures on formal power series over a finite field

For various reasons not important for this question, I'd like to show that certain subsets of $F_p[[t]]$, the ring of formal powers over the finite (prime) field $F_p$ in the variable ...
3
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1answer
129 views

Reference request - existence of formal solutions for integrable connections

Let $K$ be a field of characteristic $0$, let $A = K[[t_1, \ldots, t_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes_A \Omega^1_{A/K} ...
6
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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 ...
6
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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 ...
6
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1answer
215 views

A $q$-series identity: proof request

Denote the $q$-expressions $[q]_n=(1-q)\cdots(1-q^n)$ for $n\geq1$ and $[q]_0:=1$. Also, $[q]_{\infty}=(1-q)(1-q^2)(1-q^3)\cdots$. QUESTION. Is this identity true? It seems to be. $$\sum_{n=0}^{\...
5
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1answer
180 views

“One half of a theta-function” - is there something in the literature about it?

In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series $$ F(t,q):=\sum_{n=1}^\...
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0answers
220 views

Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?

Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
6
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1answer
321 views

Looking for infinite series resembling an exponential

I'm looking for some $f(x)$ that has the following property: $\sum_{x=1}^\infty f(kx) = r^k$ for some real $0 < r < 1$, and at least for strictly positive integer $k$. Does such an $f(x)$ ...
2
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0answers
177 views

Closed form for product of Stirling numbers of the second kind

What does the following expression evaluate to: \begin{equation} \sum\limits_{k=1}^n \dbinom{n}{k} \cdot k! \begin{Bmatrix} n \\ k \end{Bmatrix} \cdot k! \begin{Bmatrix} n \\ k \end{Bmatrix} \end{...
4
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2answers
157 views

Density of Lacunary Functions

I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...
1
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1answer
90 views

Laplace transform of the tetration (integral or series)

How to get some insight in the following integral: \begin{equation} \mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x \end{equation} where $s$ is real (and the lower integration bound may be set ...
2
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0answers
63 views

Bivariate power series as rational function

Suppose we have a bivariate power series of the form $$\sum_{i}\sum_j a_{i,j} t^i s^j,$$ where for every fixed value of $i$ the corresponding univariate power series in $s$ is a rational function. Are ...
2
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0answers
106 views

Calculating the Taylor series, given a functional equation

I have two functions, whose Taylor series about infinity are given by $$ f(z) = \frac{1}{z} + \sum_{n=2}^{\infty} \frac{A_k}{z^k}, \quad g(z)=\frac{1}{z} + \sum_{n=2}^{\infty} \frac{B_k}{z^k} $$ and ...
6
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2answers
314 views

Partitions, $q$-polynomials and generating functions

Recall the integer partition function $P(n)$ with generating function $$\sum_{n\geq0}P(n)x^n=\prod_{k=1}^{\infty}\frac1{1-x^k}.$$ Let $[n]_q=\frac{1-q^n}{1-q}$ denote the $q$-analogue of the integer $...
5
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2answers
622 views

Searching for a proof for a series identity

The below identity I have found experimentally. Question. Is this true? If so, may you provide a "slick" (or any) proof. $$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
7
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1answer
247 views

Descartes' rule of signs for infinite series

Consider the function given by $$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$ where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
3
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0answers
118 views

Cohen structure theorem with explicit equations

By Cohen structure theorem, a complete regular equicharacteristic Noetherian local ring is isomorphic to a power series. In particular, this should hold for finite extensions of power series $k[[t]][\...
-1
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2answers
330 views

Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?

Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research. There do not exist $a,b$ and $c$ such that$$ (2a-1)(2^{(b+c)}-3^c )=2^b-1 $$where $a>0,b&...
16
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1answer
441 views

Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?

I asked this question on Mathematics Stackexchange (link), but got no answer. Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$. Recall ...
4
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0answers
172 views

power series and roots of unity

Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
4
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0answers
80 views

Rationality of power series whose coefficients are the ranks of a sequence of matrices

Recently, I stumbled several times about the problem to decide whether a certain formal power series $$ f = \sum_{n=0}^\infty d_n T^n \in \mathbb{Q}[\![T]\!]$$ is actually a rational function, where ...
1
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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", ...
10
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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}...