# Questions tagged [power-series]

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318
questions

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293 views

### 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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**1**answer

479 views

### 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 ...

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155 views

### 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 $\overline{\mathbb{F}_p}((t))$ ...

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206 views

### 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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139 views

### 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) ...

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139 views

### 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 ...

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255 views

### 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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1k views

### 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 ...

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121 views

### On the remainder of a power series evaluated on the boundary of its convergence disk

Background
This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...

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131 views

### Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa?
Let's consider this method:
Transformation even->odd:
Suppose $f_{even}(x)$ is a function which satisfies the following ...

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179 views

### Does this method analytically continue gap series series?

I was looking for ways to continue gap series, and it seemed to be that they could be continued outside of the boundary by simply turning
$$f(x)= \sum_{n=0}^\infty x^{n^k}$$
into
$$g(x) =- \sum_{n=1}^\...

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77 views

### Requesting proof of closed form of sum involving Fibonacci and Lucas numbers

$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}...

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416 views

### In search for a counterexample related to the Abel-Stolz theorem

Disclaimer: I posted this question seven days ago here on the Math.SE, with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so ...

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91 views

### 'Partial boundedness' of continuously parametrised power series

Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...

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**1**answer

101 views

### Small power series "approximating" a Dirac

Does there exist a (sequence of) power series $\sum_{i\geq 0} a_{n,i} x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_{i\geq 0} \vert a_{n,i}\vert n^i=O(n^p)$ for some ...

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55 views

### Convergence of Hahn series

Enumerate $\Bbb{Q}^+$ with $\Bbb{Z}^+$ by a bijiective map $f:\Bbb{Z}^+ \rightarrow \Bbb{Q}^+$. Consider the Hahn series: $$P_f(x)=\sum_{n=1}^{+\infty}c_nx^{f(n)}$$ where $c_n \in \Bbb{C}$, $x \in \...

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24 views

### Series solution of an ODE with nonpolynomial coefficients

Basically, I have a second-order differential equation for $g(y)$ and I want to obtain a series solution at $y=\infty$ where $g(y)$ should vanish. That would be easy if the ODE contains polynomial ...

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265 views

### Finding the summation formula for the recurrence relation $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$

The exponential generating function of this recurrence relation, $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$, is
$$f(x)=e^{x + \frac{x^2}{2}}$$
Multiplying the exponential generating functions for each term, ...

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88 views

### Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...

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100 views

### Convergence radius of double series with Pochhammer symbols

I would like to know the convergence radius of the following two double power series of $(x,y) \in \mathbb{C}^2$:
\begin{align}
\sum_{m,n=0}^\infty \frac{(d-a)_{n+m}(d+b)_{n+m}(d+a)_n(d-b)_n}{n!m!(2d-...

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175 views

### Taylor's theorem for embedded manifolds

For an embedded Riemannian manifold $M \subseteq \mathbb{R}^m$ and a point $x \in M$, there is a series expansion (page 8 of Monera's paper):
$$\exp_x(t v) = x + t J_x(v) + \frac{t^2}{2!} Q_x(v) + \...

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129 views

### A *natural* polynomial expansion of the Riemann $\xi(s)$ function

This is something that I've known for some time. Its an expansion of Riemann's $\xi$ function based on the traditional representation
$$
\xi(s)=\frac{1}{2}\left(1-s(1-s)\int_{1}^{\infty}\frac{\psi(x)}{...

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41 views

### Is it possible that in certain rings the power series representing special functions are expressable via series representing elementary functions?

Let's consider the Taylor power series of a function on real numbers.
Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be ...

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139 views

### Is a mixture of real analytic functions again analytic?

Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$
Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$.
Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that
$$...

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89 views

### Request for bibliographic information

Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding:
...

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**1**answer

291 views

### Analytic functions in arbitrary rings?

We have developed a rich theory of analytic functions over $\mathbb{R}^n$ and $\mathbb{C}^n$. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to ...

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57 views

### Confluent heun function and its expansion around x=0 and x=1

Let us consider the confluent Heun equation (CHE) on the domain $D=(0, +\infty)$ in its non-symmetrical canonical form:
$H''(x)+ (\alpha+ \frac{\beta+1}{x}+\frac{\gamma+1}{x-1})H'(x)+ (\frac{\sigma}{...

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61 views

### Completion of $K$-algebra of finite type with respect to the residue norm

Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, ...

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51 views

### Approximate identities on the unit disk and going beyond a power series' radius of convergence

Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...

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85 views

### How to define Weierstrass degree of higher dimensional $p$-adic power series?

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integer $O_K$ and unique maximal ideal $m_K$.
If $f(x)$ be a power series in $O_K[[x]]$, then the Weierstrass degree of $f(x)$ denoted ...

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121 views

### What about zeros of higher dimensional power series?

Let $K$ be a p-adic field, $O$ be the ring of integers and let $m$ be its maximal ideal. Let $f(X)=(f_1(X),~f_2(X), \cdots, f_n(X)) \in O[[X]]^n,\ X=(x_1,~x_2, \cdots, x_n)$ is a power series.
I am ...

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106 views

### When can we decompose a multivariable p-adic power series into product of single variable power series?

Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?
For example, consider the following power series in $n$ variables:...

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**1**answer

208 views

### Sum of the geometrico-factorial series

How can I find the sum of the series
$$ 1+1x + 2! \cdot x^2 + 3!\cdot x^3 + \cdots + n! \cdot x^n $$
I was solving this just out of fun but now it doesn't give away. How to form a general formula for ...

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**0**answers

130 views

### Infinite sum in power series ring

Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...

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133 views

### A series that is algebraic?

This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic ...

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828 views

### A series that is rational?

Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no ...

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71 views

### Subalgebras of a polynomial ring carved out by (families of) coefficient equalities

Let $\mathbf{k}$ be a field, and let $P=\mathbf{k}\left[ x_{1},x_{2}
,\ldots,x_{n}\right] $ be a polynomial ring over $\mathbf{k}$ in $n$
variables $x_{1},x_{2},\ldots,x_{n}$. Alternatively, $P$ can ...

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**1**answer

420 views

### Approximating power series coefficients --- Why does a clearly illegitimate method (sometimes) work so well?

For reasons that don't matter here,
I want to estimate the power series coefficients
$t_{ij}$ for the rational function
$$T(x,y)= {(1+x)(1+y)\over 1- x y(2+x+y+x y)}=\sum_{i,j} t_{ij}x^iy^j$$
Using a ...

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74 views

### Minimizing coefficients in a product related to the Rogers Ramanujan identity

Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$:
$(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$...
Now replace some of the ...

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1k views

### Is this formal noncommutative power series identity known?

I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...

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**1**answer

453 views

### Partial sums of $\sum_0^\infty z^n$

Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma_z$ the set of all numbers ...

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206 views

### Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?

Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...

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**1**answer

68 views

### Can this function be interpolated with a small power series

Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$?
I feel the answer might be no but I'm not ...

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**1**answer

172 views

### Smoothness of the radius of convergence

Let $(x\mapsto a_n(x))_n$ be a sequence of smooth functions defined on some fixed interval $I$. Consider the power series $\sum_{n\geq 0}a_n(x)t^n$ and denote by $R(x)$ its radius of convergence. Does ...

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63 views

### Fixing constants of a series solution of a fourth-order PDE

The following is the PDE I want to solve,
$$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}...

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326 views

### Weak version of Karamata's Tauberian theorem

I first posted this on mathematics. However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought.
Karamata's Tauberian theorem states the following. ...

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37 views

### Proving that a quotient of hypergeometric functions is smaller than a certain function

Im trying to prove that $\forall w \in (0,1), \forall k \in \left(0,\frac{1}{5}\right)$:
$$h_k(w) = \left[\frac{_2F_1\left(\frac{3}{2},1+\frac{1}{k};\frac{1}{2}+\frac{1}{k};\frac{1-w}{1+w} \right) }{...

**43**

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**5**answers

2k views

### An "analytic continuation" of power series coefficients

Cauchy residue theorem tells us that for a function
$$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$
the coefficient $a(k)$ can be extracted by an integral formula
$$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...

**1**

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**2**answers

131 views

### Integral expressions for Bessel-like power series

I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\...

**2**

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**1**answer

268 views

### Are there any necessary conditions of lacunary functions known?

On the internet, most theorems about lacunary function only give the sufficient conditions. For example, Ostrowski-Hadamard Gap Theorem concerns the asymptotic length of null Taylor coefficients, ...