Questions tagged [power-series]

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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 ?
Jaime Yerbabuena's user avatar
6 votes
2 answers
306 views

Infinite sum of even Bessel functions - Identities

Recently, I came across the following identities among first-kind Bessel functions, namely $$ 2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1} $$...
Alessandro Pini's user avatar
8 votes
1 answer
584 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 ...
Rishabh Kothary's user avatar
6 votes
0 answers
192 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 ...
Sam Hopkins's user avatar
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3 votes
1 answer
285 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 \...
Pietro Majer's user avatar
  • 56.3k
3 votes
1 answer
268 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 ...
qifeng618's user avatar
  • 828
7 votes
1 answer
456 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 ...
Daniele Tampieri's user avatar
14 votes
1 answer
676 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, ...
user115748's user avatar
3 votes
1 answer
111 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 ...
rimu's user avatar
  • 749
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}) = ...
Rishabh Kothary's user avatar
1 vote
1 answer
105 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 ...
Rishabh Kothary's user avatar
2 votes
2 answers
196 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 ...
qifeng618's user avatar
  • 828
1 vote
1 answer
112 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 ...
joaopa's user avatar
  • 3,657
4 votes
2 answers
574 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 ...
total dependent random choice's user avatar
3 votes
1 answer
357 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 ...
L.L's user avatar
  • 399
7 votes
1 answer
509 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 ...
Amr's user avatar
  • 1,025
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 ...
IscoBerlin's user avatar
2 votes
1 answer
183 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. ...
Amr's user avatar
  • 1,025
6 votes
1 answer
240 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-...
KJL's user avatar
  • 113
4 votes
2 answers
329 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 $\...
XL _At_Here_There's user avatar
0 votes
0 answers
84 views

Boundaries on sum of digits of powers of first n natural numbers

In the following, I'm presenting my work, Boundaries on sum of digits of power of first n natural numbers. Let function $D(x,y)$ shows sum of digits of $y$ in base $x\ge2$. Example $D(10,234)=2+3+4=9$ ...
Pruthviraj's user avatar
0 votes
0 answers
90 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}{...
Libli's user avatar
  • 7,210
1 vote
0 answers
46 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 ...
qifeng618's user avatar
  • 828
4 votes
4 answers
629 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+\...
qifeng618's user avatar
  • 828
2 votes
1 answer
128 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 ...
TheStudent's user avatar
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]]...
Alexander Burstein's user avatar
-2 votes
1 answer
203 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(...
MrPie 's user avatar
  • 185
2 votes
2 answers
326 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 \...
Ruy's user avatar
  • 2,233
2 votes
0 answers
180 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 ...
esner1994's user avatar
2 votes
1 answer
134 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) ...
qifeng618's user avatar
  • 828
2 votes
1 answer
249 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 ...
Daniele Tampieri's user avatar
2 votes
1 answer
69 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 $\...
Andi Bauer's user avatar
  • 2,901
4 votes
1 answer
149 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 ...
Sanae Kochiya's user avatar
2 votes
1 answer
96 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 $K r$, $$ r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta $$ ...
apg's user avatar
  • 612
3 votes
0 answers
107 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 ...
Dror Speiser's user avatar
  • 4,563
4 votes
1 answer
206 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 ...
M masa's user avatar
  • 479
3 votes
1 answer
320 views

A few reference questions about the Baker–Campbell–Hausdorff formula

I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions. Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$ $$ \...
askquestions2's user avatar
-1 votes
1 answer
95 views

A proof of an interesting inequality [closed]

If $0<\beta<1$ and $0<x<1,$ how to prove that $$h(x)-2x+(4-2^{1+\beta})x^{1+\beta}<0,$$ where $$h(x)=(1+x)^{1+\beta}-x^{1+\beta}-1.$$The numerical simulation shows that it is true.
Renjun Qi's user avatar
11 votes
1 answer
930 views

In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
Madeleine Birchfield's user avatar
1 vote
1 answer
140 views

Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?

Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$. Let $R_K$ be ring of integers of $K$. Let $ \hat{E}$ be its formal group of $E$. Take $...
Duality's user avatar
  • 1,397
1 vote
0 answers
167 views

Homomorphism of formal group of elliptic curve corresponding to its endomorphism

Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group. Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
Duality's user avatar
  • 1,397
1 vote
0 answers
143 views

What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
Raphael B's user avatar
7 votes
0 answers
156 views

Poincaré series of deloopings of finite complexes

Recall that the Poincaré series of a topological space $X$ is defined as $P_X(t) = \sum_{j=0}^{\infty} b_jt^j$, where $b_j = \text{dim}_{\mathbb Q} H_{j}(X;\mathbb Q)$ means the $j^{\text{th}}$ (...
Jens Reinhold's user avatar
3 votes
0 answers
181 views

Reshuffling power series (aka Melvin–Morton expansion in knot theory)

I am struggling to understand a statement which follows from some change of variables in a power series. I think that the context does not really matter here, so I will put it at the end of the ...
Minkowski's user avatar
  • 571
3 votes
1 answer
301 views

Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as $$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\...
Raphael B's user avatar
0 votes
1 answer
93 views

Explicit expression of Padé–Hermite approximant of type I

It is well known that the Padé approximants $(P,Q)$ of an analytic function in the neighborhood of $0$ can be expressed as a quotient of Hankel determinants built on the coefficients of the function $...
joaopa's user avatar
  • 3,657
9 votes
2 answers
716 views

Algebraic power series of finite order

Apologies if the question is too elementary/something well-known. I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
Sam Hopkins's user avatar
  • 22.5k
0 votes
1 answer
144 views

Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?

This question is based on this Math.SE answer, so let's recall a few concepts dealt with there. If $\{a_n\}_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum_{n=0}^\infty a_nz^n$ ...
Daniele Tampieri's user avatar
9 votes
1 answer
394 views

Linear recurrence relation for symmetric powers in the Burnside ring

Let $G$ be a finite group and $B(G)$ be its Burnside ring, i.e. formal sums of isomorphism classes of finite $G$-sets with addition given by disjoint union and multiplication given by Cartesian ...
Aleksei Piskunov's user avatar
3 votes
0 answers
115 views

On tangential approach regions for general power series converging on the unit disk

Notation and premises. Here it is a list of notations more or less explicitly used in the question: If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...
Daniele Tampieri's user avatar

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