# Questions tagged [power-series]

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286
questions

**0**

votes

**1**answer

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**

votes

**0**answers

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 $A$. ...

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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

**1**answer

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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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

**1**answer

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 ...

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**0**answers

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**

votes

**1**answer

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

**1**answer

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

**1**answer

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 ...

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votes

**1**answer

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

**1**answer

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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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}...

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votes

**2**answers

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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**0**answers

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**

votes

**2**answers

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

**1**answer

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

**1**answer

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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**0**answers

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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**1**answer

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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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**

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**1**answer

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

**0**answers

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**

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**1**answer

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**

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**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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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**0**answers

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

**1**answer

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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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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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**

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**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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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**0**answers

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
$f(n)=n$. ...

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vote

**0**answers

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

**1**answer

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**

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**2**answers

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**

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**1**answer

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\...

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vote

**0**answers

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

**0**answers

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**

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**0**answers

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**

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**1**answer

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

**2**answers

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**

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**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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$\colon$
$$
f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.
$$
...