Questions tagged [power-series]

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votes
1answer
151 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
100 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 $A$. ...
1
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0answers
126 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
votes
1answer
806 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 ...
4
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0answers
54 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 ...
10
votes
1answer
333 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 ...
0
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0answers
66 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 ...
25
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1answer
765 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
votes
1answer
429 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 ...
-4
votes
1answer
193 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
66 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
votes
1answer
143 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
62 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
votes
2answers
264 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
33 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) }{...
20
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2answers
665 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
vote
1answer
88 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 $\...
3
votes
1answer
177 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, ...
4
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0answers
130 views

Nascent formal group law

The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal compositional inverse, perhaps ...
-3
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1answer
131 views

A proposition about power series

Is this proposition established? Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series $$p(x)=\sum_{n=0}^\infty a_nx^n,$$ $$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+...
3
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0answers
113 views

Rational functions and power series

Let $P(z)/Q(z)$ be a rational function ($\mathbf{C}$ coefficients) and assume that its Taylor series $\alpha(z)=\sum_{n\geq 0} a_n z^n$ around $z=0$ has radius of convergence $1$. Consider the power ...
12
votes
1answer
781 views

Singularities of power series

The power series $\sum_{n=1}^\infty \ln(n)z^n$ has radius of convergence $1$ and $z=1$ is a singular point. Is $z=1$ an isolated singularity? If yes, what kind of isolated singularity? I am only able ...
2
votes
0answers
102 views

Multivariate Weierstrass preparation Theorem?

Let $(K,|\cdot|)$ be a complete local field and $\mathcal{O}$ be its ring of integers. Let $C$ be a complete algebraic closure of $K$ and let $\mathfrak{m}:=\{x\in \mathcal{O}_{C}~|~|x|<1\}$ where $...
2
votes
1answer
87 views

A question relating to certain algebraic manipulation of a formal power series written in the form of infinite product

Suppose there is formal power series in infinite product form as follows: $$\prod_{d\geq 1} \left(1+\frac{u^d}{q^d-1}\right)^{a_d}$$, where $a_d$ are positive integers. Consider the expression $$\...
3
votes
1answer
129 views

Proving that a morphism between power series rings is regular

Let $k\subset K$ be a separable field extension. As a particular case of M. André Localisation de la lissité formelle one obtains that the natural inclusion of power series rings $k[[X_1,\ldots,X_n]]\...
4
votes
2answers
345 views

Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$

Can one characterize the $a\in\mathbb F_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer. Thanks in advance for any help.
5
votes
1answer
370 views

Convergence of the series of Legendre polynomials

Consider the generating function of Legendre polynomials: $$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$ Is it true that for $0<x<1, t=1$ series of Legendre ...
1
vote
1answer
73 views

Intersection of a certain linear ideals of $K[[X_1,\ldots,X_{np}]]$ for ${\mathrm{ch}}(K) = p > 0$

Suppose ${\mathrm{ch}}(K) = p > 0$ and we consider the formal power series ring $K[[X_1,\ldots,X_{np}]]$ over $K$ in $np$ variables $X_1,\ldots, X_{np}$. Let $\Lambda$ be the set defined as follows$...
2
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0answers
28 views

The total Wronskian

Given a sequence of function, $$F=\{f_1(x,t),f_2(x,t),f_3(x,t),\cdots,f_m(x,t)\},$$ we define the total Wronski determinant of this set of functions as $$W(F)=\det\begin{vmatrix}F\\D_xF\\D_tF\\\vdots\\...
2
votes
1answer
286 views

An element of formal power series over a commutative ring

Let $R$ be a commutative ring with 1 and let $R[[x]]$ be the formal power series ring over $R$. Now let $f\in R[[x]]$ with the property that if $g, h\in R[[x]]$ and $f=g+h$ then either $\langle h \...
1
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0answers
104 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
128 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 ...
3
votes
0answers
74 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
votes
1answer
192 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
votes
0answers
40 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
votes
1answer
332 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
105 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 $f(n)=n$. ...
1
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0answers
172 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
votes
1answer
247 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
votes
2answers
294 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
votes
1answer
241 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
203 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 ...
1
vote
0answers
38 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
votes
0answers
125 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}...
0
votes
1answer
111 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
votes
2answers
192 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
votes
2answers
495 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
votes
2answers
396 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
votes
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
votes
1answer
116 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$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...

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