# Questions tagged [power-series]

The power-series tag has no usage guidance.

355
questions

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### A proof of an interesting inequality

If $0<\beta<1$ and $0<x<1,$ how to prove that $$h(x)-2x+(4-2^{1+\beta})x^{1+\beta}<0,$$ where $$h(x)=(1+x)^{1+\beta}-x^{1+\beta}-1.$$The numerical simulation shows that it is true.

10
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1
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694
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### In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph:
Someone mentioned (I think on Twitter) that the Taylor ...

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0
answers

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### Criterion to decide whether a function is algebraic

For Christol's theorem see 1, if a power series is algebraic over every fields of characteristics $p$, is it algebraic over fields of $0$, by Robinson's principle?
Update: The power series is in $\...

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votes

1
answer

92
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### Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?

Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.
Take $...

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0
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146
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### Homomorphism of formal group of elliptic curve corresponding to its endomorphism

Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group.
Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads
For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...

1
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0
answers

122
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### What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...

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140
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### Poincaré series of deloopings of finite complexes

Recall that the Poincaré series of a topological space $X$ is defined as $P_X(t) = \sum_{j=0}^{\infty} b_jt^j$, where $b_j = \text{dim}_{\mathbb Q} H_{j}(X;\mathbb Q)$ means the $j^{\text{th}}$ (...

3
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### Reshuffling power series (aka Melvin–Morton expansion in knot theory)

I am struggling to understand a statement which follows from some change of variables in a power series. I think that the context does not really matter here, so I will put it at the end of the ...

3
votes

1
answer

278
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### Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\
& a_{n,m} \geq 0 \quad \forall n, m \in\...

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0
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### Explicit expression of Padé–Hermite approximant of type I

It is well known that the Padé approximants $(P,Q)$ of an analytic function in the neighborhood of $0$ can be expressed as a quotient of Hankel determinants built on the coefficients of the function $...

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2
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602
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### Algebraic power series of finite order

Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...

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1
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### Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?

This question is based on this Math.SE answer, so let's recall a few concepts dealt with there. If $\{a_n\}_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum_{n=0}^\infty a_nz^n$ ...

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353
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### Linear recurrence relation for symmetric powers in the Burnside ring

Let $G$ be a finite group and $B(G)$ be its Burnside ring, i.e. formal sums of isomorphism classes of finite $G$-sets with addition given by disjoint union and multiplication given by Cartesian ...

3
votes

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### On tangential approach regions for general power series converging on the unit disk

Notation and premises. Here it is a list of notations more or less explicitly used in the question:
If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...

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### Can one find an elementary function $f(t)$ such that ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)=f(t)$?

For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}_2F_1(\alpha,\beta;\gamma;z)$ can be defined by the series
\begin{equation}\...

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votes

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### Faster computation of p-adic log

As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting).
When it comes to computing $\log P(x)$, one may use the formula
$$
(\log P)' = \...

0
votes

0
answers

66
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### Which power series in $\mathbb{Z}_p[[T]]$ are rational functions? [duplicate]

Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as:
$$f(...

2
votes

1
answer

260
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### What's the fastest way to compute $\log n$ for $n>1$?

As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series
$$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$
Thus, to compute $\log n$ with $n>1$, we may employ the ...

0
votes

0
answers

127
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### Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms

Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...

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195
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### A conjectural identity involving infinite series

Recently I formulated the following curious conjecture based on my computation.
Conjecture. For all $|x|>1$, we have the identity
$$\sum_{k=0}^\infty\frac{\sum_{j=0}^{k}\binom{2k+1}{2j}(1-x)^jx^{k-...

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### Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...

5
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1
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237
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### Is the minimal polynomial of an algebraic formal Laurent series always separable?

Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from ...

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0
answers

97
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### Property of an integer sequence related to series reversion

Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:
Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...

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1
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249
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### Are algebraic power series in positive characteristics D-finite?

We know that in characteristic $0$, all algebraic series are differentiably finite.
Is this true in positive characteristic? I look at the proof, indeed we need to the
characteristic to be $0$ for the ...

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vote

2
answers

251
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### Power series of ratio of Gamma functions

Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...

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### Is there an infinite combinatorics of common transseries expansions?

By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...

2
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0
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### Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$

I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...

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0
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### Maximum likelihood estimator for power law with negative exponent

Background
I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration):
Now I ...

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0
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### What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?

Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.
Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...

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### Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$

Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true ...

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### How to prove this equation? [closed]

How to prove this equation:
\begin{align}\sum_{k=1}^{n}\cos ^{2m+1} \frac{(2k-1)\pi}{2n+1} =\frac{1}{2}\end{align}
where $k,m,n\in \mathbb{N}^*$.

0
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1
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### Levi-Civita field in unusual basis

Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}...

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2
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### In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?

In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
Particularly, is there an element $w$ of the field such that the ...

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0
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### Generalization of Levi-Civita type construction towards divergent integrals and corresponding questions

A known generalization of Levi-Civita field is a field of Hahn power series of $\varepsilon$ of the form
$\mathbb{R}[[\varepsilon^{\mathbb{Q}}]]$. Assuming $\varepsilon=1/\omega$, we can naturally ...

7
votes

1
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365
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### Improving Cauchy estimates?

Consider an entire function $f:\mathbb C \to \mathbb C$ that is real on the real line and even.
This function has a Taylor series of the form
$$f(z) = \sum_{i=0}^{\infty} a_i z^{2i} \text{ with } a_i \...

10
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2
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447
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### Lagrange inversion for power-series with rational powers

One can use Lagrange inversion to find the
power series $F(x)$, which solves $F(x) = x(1+F(x)^p)$,
where $p$ is a positive integer.
Now, what if $p$ is not an integer, but rather a positive rational ...

5
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1
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### 'Lie correspondence' for formal power series in non-commuting indeterminates

This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...

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1
answer

334
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### Closed form series for reciprocal cubic function

consider a cubic of the form f(x)=$x^3-2x+z$
Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...

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1
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### Power series whose coefficients are limits of coefficients of polynomial interpolations

When can you reconstruct the power series of a function by taking the limits of the coefficients of the polynomials that interpolate its values at $0,1,2,\dots,j$?
More precisely:
Let $f\colon\mathbb{...

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0
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### Eisenstein theorem for algebraic power series in positive characteristic

Eisenstein proved that if a power series $\sum_{n\ge0}a_nz^n$ over $\mathbb C$ is algebraic over $\overline{\mathbb Q}(z)$, then it exists positive integers $a$ and $b$ such that for all $n\in\mathbb ...

3
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1
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207
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### Two questions about Bell polynomials

In my research in Quantum Field Theory, I have encountered two questions that involve partial Bell polynomials:
Let $u$ and $x_i$ be indeterminates. I have checked that the following conjectured ...

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1
answer

206
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### Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$.
...

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2
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### Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?

According to numerical simulation, the relationship
$$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$
where $\Gamma$ is the Gamma function seems to be true.
Do you ...

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757
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### Divergent series summation beyond natural boundaries

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...

1
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0
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163
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### Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ?
I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...

2
votes

0
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213
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### Is there a closed form of $ \displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$

where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...

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156
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### How to explicitly obtain an analytic function whose power series coefficients are sums over integer compositions?

Starting with the following differential equation,
\begin{eqnarray}
x \frac{\partial^3}{\partial x^3} P[h, x] - \frac{\partial^2}{\partial h^2} \left( h \frac{\partial}{\partial h} P [h, x] \right) ...

2
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0
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146
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### How could this difference in series of power of zeros associated to counting integers and counting primes be explained?

Introduction:
In this 1992 paper, J.B. Keiper (an amazing person, who tragically died way too young), derives several power series expansions of the Riemann $\xi$-function that involve infinite sums ...

6
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295
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### Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series)

The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be ...

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5
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### Comparing two power-series

I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself.
Let $f(z)$ be a ...