Questions tagged [power-series]

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2answers
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 ...
10
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1answer
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 ...
1
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0answers
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 $\overline{\mathbb{F}_p}((t))$ ...
2
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0answers
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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0answers
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) ...
2
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0answers
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 ...
6
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1answer
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 ...
15
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5answers
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 ...
5
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0answers
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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0answers
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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2answers
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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0answers
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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2answers
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 ...
1
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2answers
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 $...
3
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1answer
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 ...
4
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0answers
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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0answers
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 ...
3
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2answers
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, ...
3
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0answers
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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2answers
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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0answers
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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0answers
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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0answers
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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0answers
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: ...
4
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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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1answer
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:...
0
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1answer
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 ...
3
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0answers
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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0answers
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 ...
9
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1answer
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 ...
3
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0answers
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 ...
9
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1answer
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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0answers
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 ...
28
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1answer
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 + \...
15
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1answer
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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1answer
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 ...
0
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1answer
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 ...
4
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1answer
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 ...
1
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0answers
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}...
2
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2answers
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. ...
3
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0answers
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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5answers
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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2answers
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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1answer
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, ...

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