The power-series tag has no wiki summary.

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### Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series
$$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$
Also, a particular ...

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### Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?
More specifically:
i) What types of essential singularities can occur?
ii) Are ...

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35 views

### t-linked extension

Let $A\subseteq B$ be an extension of commutative integral domains. the extension is t-linked if it satisfies the following property:
If P is a finitely generated ideal of A such that $P^{-1}=A$ than ...

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39 views

### Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...

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433 views

### How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically.
...

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### Relative nonarchimedean disks and annuli

Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.
Is there a good notion of closed disk of ...

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842 views

### Which power series have bounded integral coefficients and have an inverse given by a series having bounded integral coefficients

Let $A=1+\sum_{n=1}^\infty \alpha_nx^n\in\mathbb Z[[x]]$ and $B=\frac{1}{A}=1+\sum_{n=1}^\infty\beta_n x^n$ two mutually inverse power series
having bounded integral coefficients (ie. $\vert ...

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329 views

### Find an integrable, positive, unbounded, analytic function

Is there a standard example of a function $f \in L^1( \mathbb R)$ which is analytic, positive, integrable but not bounded?
An example which comes immediately to mind is to take the series of narrower ...

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130 views

### The rigid-analytic open disk

Let $K$ be a local field and $D_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Proposition 1.1 of this article describes ...

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### Matching power series to infinity

As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...

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324 views

### properties of formal delta functions

The formal delta function is
$\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n.
$
If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...

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### Estimate of the sum Taylor's coefficients

Let
$f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{x}{e^x-1}, \quad x < 0. \end{cases}$
Power series in 0:
$f(x) = \sum_{n=1}^{\infty} a_n x^n = ...

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215 views

### Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function? [closed]

Given a series with integral coefficiens as following:
$$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...

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130 views

### Ordinary or Rational Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary or rational generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted ...

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61 views

### For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient?

In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”.
...

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66 views

### Orders of certain quotients of power series rings

Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible ...

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190 views

### integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting.
Let $M$ be a smooth curve over an algebraically closed field $k$. Let ...

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263 views

### How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways:
$$1 + \sum_{i=1}^\infty f_i t^i =
\prod_{i=1}^\infty (1-t^i)^{-n_i}$$
Here, the $f_i$ and $n_i$ ...

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150 views

### Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition

there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the ...

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455 views

### Combinatorial interpretation of composition of power series?

This is a minor curiosity that came up in a joint project recently.
Consider the sequence $a_n=3\frac {(2n)!}{(n+2)!(n-1)!}$ (A000245 in OEIS).
It has multiple combinatorial descriptions.
One can ...

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**1**answer

97 views

### A pole of function in the article of Springer

I read the article Springer, T.A. On the invariant theory of $SU_2$, Indag. Math. 42, 339-345 (1980). Author considers $\mathbb{C}$-linear map at page $340.$
If $n$ is a positive integer, then write ...

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209 views

### R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module.
Would $R$ noetherian ...

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288 views

### Laurent expansion of a principal value integral

Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form:
$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$
...

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214 views

### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?

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### derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has
...

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131 views

### Randomness about coefficients of series

$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if ...

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### Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!

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### An apparently simple question (behaviour at infinity of a power series)

Let $(a_n)$ be a sequence of real numbers, and suppose that the real power series (function) $S(x):=\sum_{n=0}^{\infty} a_n x^n$ converges for every $x\in\mathbb{R}$.
$\mathbf{Question}$: Suppose ...

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### Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in ...

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232 views

### On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...

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### derivatives of composite function [closed]

There's a formula for the $n$th derivative of a composite function $f(g(x))$ - it's called Faa di Bruno's formula - but I'm not really interested in the formula but in the proof given in the book of ...

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### Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer.
Over $A$ we consider the Tate-algebra
$$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...

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### Positivity of coefficients of a power series

How does one check for the positivity of coefficients of a rational function,say, for example $\frac{p_1(x,t)}{(1-xt)(1-x^2t)(1-x^3t)}$ where $p_1(x,t) = 1 + tx + 2t^2x^2 - 3x^3t^2 -x^5t^3 - ...

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### name for a degree-like invariant of a power series over a commutative ring

Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f ...

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281 views

### existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a ...

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254 views

### Exponential of a specific hypergeometric series

This is motivated by this question.
Let $f$ be the hypergeometric series
$ f(x) = 2 x \, _{4}F_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $
which is explictly given by
$ f(x) = \sum_{n \geq 1} ...

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243 views

### Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$

Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$

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### Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = ...

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172 views

### An extension of the real semiring with multiple degrees of infinity

Is it possible to define an extension of the probability semiring $(\mathbb{R}^+, +, \times, 0, 1)$ such that
Closure $a^* = 1 + a + a^2 + \ldots$ is defined for every element of the semiring, not ...

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### Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...

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### Convolution-type operator for series

Suppose $f(z)=\lambda(a_1z+a_2z^2+\cdots)$ is holomorphic in $\{|z|<1\}$ with $\lambda>0$. For each $d\geq 1$ , I am trying to define an operation, $\star_d$ , so that $f(z)\star_d ...

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244 views

### A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions:
(1+x+x$^2$+...)(1+x$^2$+x$^4$+...)(1+x$^3$+x$^6$+...)...
Now replace some of the plus signs with minus signs and expand the product into a ...

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### Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?

Let D in Z/3[[x]] be sum ((a_n)(x^n)) where the sum runs over all n prime to 6 and a_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So ...

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### What's the asymptotic behavior of this function at large distance? [closed]

This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where ...

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### Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?

This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in ...

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### elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?

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### Solving ODE with negative expansion power series

I'm moving this here, as suggested from physics.stackexchange. The original is here.
So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...

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187 views

### Is it decidable whether the support of a rational $\mathbb{Z}$-series is a regular language?

Let $S \in \mathbb{Z}\langle\langle A\rangle\rangle$ be a rational series in noncommutative variables. The support of $S$ is the set of all words $u \in A^*$ such that $(S, u) \not= 0$. It is ...

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### Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?

The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag.
NOTATION
Fix an odd prime $N$. Let
$$
...

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### Can the series $\sum\limits_{n=0}^\infty q^{F_n}$ be expressed in terms of theta functions?

Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to ...