# Questions tagged [power-series]

The power-series tag has no usage guidance.

405
questions

2
votes

1
answer

51
views

### Lower bound in recurrence relation

We have initial conditions: $n_1=2, v_1=1.$
And given recurrence relations:
$v_i=(2^{i+1}+1) v_{i-1}$
$n_i\leq 2^i n_{i-1}+2v_{i-1}$
I need to prove the below lower bound:
$$v_i≥n_i.e^{\Omega(\sqrt{\...

7
votes

0
answers

110
views

### A differential equation and recurrence related to P-partitions

I am interested in polynomials $G_n(z)$ defined by the recurrence
$$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$
for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$.
The next few values ...

1
vote

1
answer

85
views

### On analytic transcendence degree and Krull dimension for homomorphic images of power series rings

Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...

0
votes

0
answers

31
views

### The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function

The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...

0
votes

0
answers

121
views

### How to prove the convergence of the following series involving Gamma function?

Consider the following result（$d$ denotes the dimensions and $0<t<T$）
$$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq ...

4
votes

1
answer

114
views

### Reference request for Bessel function of the second kind with matrix argument

As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...

-4
votes

1
answer

190
views

### How to express a quadratic polynomial exactly as a power series [closed]

I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...

2
votes

0
answers

53
views

### Transform connecting powers of integration and differentiation operators

Just by a chance, I found the following power series identity, which holds for any analytic function $F(\cdot)$, nonnegative integer $m$, and constants $u,v$ not depending on indeterminates $z,t$:
$$\...

1
vote

1
answer

240
views

### Confusion regarding $\ln \omega$

This answer says that in surreal numbers $\ln \omega=\omega^{1/\omega}$.
At the same time, this Wikipedia article says that transseries $\mathbb{T}^{LE}$ are isomorphic to a subfield of $No$ with its ...

20
votes

1
answer

2k
views

### Does every series of hyperreal numbers converge to some hyperreal number?

I am currently trying to find some field $F$ which includes $\mathbb{R}$ (or $\mathbb{C}$) and in which series $x^* = \sum_{i\in\mathbb{N}} x_i$ converge to some element of the field. (i.e. $x^* \in ...

1
vote

1
answer

149
views

### Convergence of a product in $\mathbb Q_2[[X]]$

I thought it would be very easy to prove, but in fact, I did not manage to prove or disprove this fact:
the sequence of polynomials $$\left(\prod_{j=0}^k\big(1-2^{2^j}X\big)\right)_{k\in\mathbb N}$$
...

0
votes

0
answers

100
views

### Does the following series converge? To which value?

Given $f(x)=A \cdot x^5+B \cdot x^8$
with:
$A \in \mathbb{R}^-$
$B \in \mathbb{R}^-$
$h(z) = w_0 + \sum_{n=1}^\infty h_n \cdot \frac{\left(z-f(w_0)\right)^n}{n!}$
$w_0=1$
and:
$$h_n=\lim_{w \...

2
votes

5
answers

531
views

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

Can we find $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$, assuming $\{x_i\}_{i\in\mathbb{N}}$ is a set of positive real numbers?
Perhaps an easier question is, can we find $\sum_i x_i$ ...

6
votes

1
answer

259
views

### Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...

2
votes

0
answers

74
views

### Limit of lacunar power series at $1^-$

I've asked this question on MSE but I didn't get a convincive answer so I'm trying here. Here is the question :
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider ...

1
vote

0
answers

31
views

### Exponential-like function equivalent for the Dixonian Elliptics

Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?

6
votes

2
answers

473
views

### Infinite sum of even Bessel functions - Identities

Recently, I came across the following identities among first-kind Bessel functions, namely
$$
2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1}
$$...

8
votes

1
answer

624
views

### A question on algebraic independence

Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...

6
votes

0
answers

196
views

### Filling in some missing squares for classes of power series

This question concerns various important classes of formal power series. For concreteness and convenience, let us work with power series $F(x) = \sum_{n\geq 0}c_n x^n \in \mathbb{C}[[x]]$, i.e., with ...

3
votes

1
answer

289
views

### Does this condition characterise intervals, among subsets of the real line?

For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...

3
votes

2
answers

381
views

### Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...

7
votes

1
answer

474
views

### On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau

To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...

14
votes

1
answer

681
views

### Power series $x^n/n$ with one plus, then two minuses, then three plusses, and so on

Which function is represented by the powerseries
$$1-\frac{x}{2}-\frac{x^2}{3}+\frac{x^3}{4}+\frac{x^4}{5}+\frac{x^5}{6}-\frac{x^6}{7}-\frac{x^7}{8}-\frac{x^8}{9}-\frac{x^9}{10} + \dots$$
(one plus, ...

4
votes

1
answer

142
views

### Is there a generalisation of the Vivanti-Pringsheim theorem for several variables?

The Vivanti-Pringsheim theorem states that if $f(z)$ has a power series with non-negative coefficients and a radius of convergence $R > 0$, then it has a singularity at $R$. So to find the radius ...

0
votes

0
answers

109
views

### Characterisation of even characteristic quadratic system

$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that
$f_1(\bar{x}) = x_1 + x_5^2 + q_1$,
$f_2(\bar{x}) = x_2 + x_1^2 + q_2$,
$f_3(\bar{x}) = ...

1
vote

1
answer

113
views

### A question on classification of quadratic polynomials in even characteristic

$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...

2
votes

2
answers

203
views

### An identity for the ratio of two partial Bell polynomials

Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...

1
vote

1
answer

120
views

### Completion of $\mathbb F_q(T)$

It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...

4
votes

2
answers

592
views

### Reference for group-algebra/exp-log like identites in combinatorics

I've encountered several identities in combinatorics that resemble inversion formulas, as shown below,
Here, $f_i, g_k$, $\forall i,k \in \mathbb{N}$, are coefficients of some formal power series.
I ...

3
votes

1
answer

370
views

### Convergence of a power series

Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...

7
votes

1
answer

587
views

### Composition of power series is power series?

$\DeclareMathOperator\dom{dom}$Sorry to bother the community again with these type of questions about power series, I am ready to delete the question if it is not suitable.
Definition: I say a ...

2
votes

1
answer

156
views

### Matrix inequalities in series form

While studying the positivity of mechanical systems, we land on the following conjecture but don't know how to prove this.
If a square matrix $A \in \mathbb{R}^{m\times m}$ satisfies both the two ...

2
votes

1
answer

185
views

### Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...

6
votes

1
answer

251
views

### Intuitive explanations of the Carlitz-Scoville-Vaughan theorem

Crossposted from MSE:
I recently came across Ira Gessel's slides on a theorem he says should "be considered one of the fundamental theorems of enumerative combinatorics."
The Carlitz-...

4
votes

2
answers

332
views

### Is the value of the power series at 0.1 transcendental?

Let $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ where $a_n\in \{0,1\}$, and the $f(x)$ has a natural boundary. By the way, $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ is rational function or transcendental one on $\...

0
votes

0
answers

101
views

### Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...

1
vote

0
answers

48
views

### A problem on monotonicity rule for the ratio of two Maclaurin power series

In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.
Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and
the power series ...

4
votes

4
answers

675
views

### What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...

2
votes

1
answer

141
views

### Proof of Szegö asymptotic theorem

Consider the truncated exponential series
$$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$
The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...

3
votes

0
answers

69
views

### Degree of an even/odd part of a formal power series over a polynomial ring

Let $K$ be a field with $\operatorname{char}K\ne 2$ (say, $K=\mathbb{R}$ or $\mathbb{C}$) and consider a formal power series $f=f(x)\in K[[x]]$ such that $[K[x,f]:K[x]\,]=d$. Suppose $f_e,f_o\in K[[x]]...

-2
votes

1
answer

211
views

### Convergence and roots of alternating periodic infinite series

Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...

2
votes

2
answers

329
views

### When does $\lim_{s\to 1_-} (1-s)\sum_{n=0}^\infty a_ns^n$ exist?

Let $a=\{a_n\}_{n\geq 0}$ be a sequence of positive real numbers with $a_n\leq 1$, for all $n$, and observe that, for
any real number $s\in [0,1)$, one has that
$$
\sum_{n=0}^\infty a_ns^n \leq \...

2
votes

0
answers

225
views

### Power series of the modified Bessel function of the second kind

I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...

2
votes

1
answer

151
views

### Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions

Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...

2
votes

1
answer

255
views

### On a lemma of Łojasiewicz in complex analysis of one variable

Context. The question arises from my former question on the remainder of a power series. Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper ...

3
votes

1
answer

79
views

### Exponential taylor series for multiple variables with linear constraints for coefficients

I'm trying to simplify the sum
$$
\sum_{\vec x \in (\mathbb{N}_0)^n: M\vec x = \vec b} \prod_i \frac{(a_i)^{x_i}}{x_i!},
$$
where $M$ is a $\mathbb{N}_0$-valued $m\times n$ matrix, $\vec b$ is $\...

4
votes

1
answer

152
views

### Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?

Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...

2
votes

1
answer

106
views

### Power series expansion of the order parameter in the Kuramoto model

In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...

3
votes

0
answers

113
views

### Zeros of inverse of dilogarithm

I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also ...

4
votes

1
answer

209
views

### Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$

Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series ...