Questions tagged [power-series]
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391
questions
3
votes
2
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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 ...
0
votes
1
answer
385
views
Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
31
votes
3
answers
1k
views
Which power series have bounded integral coefficients and have an inverse given by a series having bounded integral coefficients
Let $A=1+\sum_{n=1}^\infty \alpha_nx^n\in\mathbb Z[[x]]$ and $B=\frac{1}{A}=1+\sum_{n=1}^\infty\beta_n x^n$ two mutually inverse power series
having bounded integral coefficients (ie. $\vert \alpha_n\...
1
vote
0
answers
26
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
313
views
Infinite sum of even Bessel functions - Identities
Recently, I came across the following identities among first-kind Bessel functions, namely
$$
2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1}
$$...
6
votes
2
answers
396
views
What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$?
$\delta$ is a positive number. Is this Taylor expansion of some function?
8
votes
1
answer
589
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
193
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
286
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 \...
7
votes
1
answer
459
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
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, ...
3
votes
1
answer
180
views
Eisenstein theorem for algebraic power series in positive characteristic
Eisenstein proved that if a power series $\sum_{n\ge0}a_nz^n$ over $\mathbb C$ is algebraic over $\overline{\mathbb Q}(z)$, then it exists positive integers $a$ and $b$ such that for all $n\in\mathbb ...
0
votes
1
answer
94
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 $...
3
votes
1
answer
115
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
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 ...
7
votes
1
answer
150
views
Is there an infinite combinatorics of common transseries expansions?
By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
3
votes
2
answers
1k
views
Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
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 ...
1
vote
1
answer
114
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
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 ...
54
votes
8
answers
9k
views
Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
4
votes
1
answer
496
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, ...
3
votes
1
answer
359
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
523
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 ...
19
votes
2
answers
744
views
Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?
I asked this question on Mathematics Stackexchange (link), but got no answer.
Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall ...
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
241
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-...
3
votes
1
answer
475
views
Relation between coefficients of expansions
Related to Relations between coefficients of expansions of a rational function at 0 and infinity
I commented at the linked question that the question seemed less about what happened "at infinity", ...
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 $\...
0
votes
0
answers
85
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 $D(10,234)=2+3+4=9$
...
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 ...
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}{...
4
votes
4
answers
634
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+\...
7
votes
2
answers
943
views
Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?
For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}_2F_1(\alpha,\beta;\gamma;z)$ can be defined by the series
\begin{equation}\...
2
votes
1
answer
130
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
2
answers
327
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
$$
\sum_{n=0}^\infty a_ns^n \leq \...
-2
votes
1
answer
203
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
2
votes
0
answers
184
views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...
2
votes
1
answer
134
views
Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions
Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...
2
votes
1
answer
250
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 ...
7
votes
1
answer
403
views
Hadamard product of specific type of power series
I am consider the power series of the form $$F_n(t):=\frac{1}{\prod_{i=1}^n(1-t^i)}.$$ Given two power series $A(t)=\sum_{i\ge 0}{a_it^i}$ and $B(t)=\sum_{i\ge0}{b_it^i}$, their Hadamard product is ...
4
votes
2
answers
790
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 ...
2
votes
1
answer
70
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
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 ...
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 $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...
4
votes
1
answer
207
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 ...
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 ...