Questions tagged [power-series]
The power-series tag has no usage guidance.
380
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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 ...
- 149
3
votes
1
answer
314
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 ...
- 389
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1
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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 ...
- 1,025
2
votes
1
answer
146
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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 ...
- 33
2
votes
1
answer
178
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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. ...
- 1,025
5
votes
1
answer
199
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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-...
- 103
2
votes
0
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115
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Embedding $K((X,Y))$ into $K((X))((Y))$
This question is closely related to Explicit elements of $K((x))((y))∖K((x,y))$.
Recall that $K((X,Y))$ can be embeded into $K((X))((Y))$, althouth this embedding is not surjective. A natural question ...
- 527
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2
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310
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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 $\...
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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 $D(10,234)=2+3+4=9$
...
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0
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81
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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}{...
- 6,960
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0
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43
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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 ...
- 716
4
votes
4
answers
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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+\...
- 716
2
votes
1
answer
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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 ...
- 907
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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]]...
- 1,078
-2
votes
1
answer
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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(...
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2
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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
$$
\sum_{n=0}^\infty a_ns^n \leq \...
- 2,183
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0
answers
115
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Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...
- 83
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0
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100
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Solving a functional equation involving the exponential generating function of the Stirling numbers of the first kind
Let $F(x)=\sum_{n\geq 2m-1} f_n \frac{x^n}{n!}$, and $F_o=\sum_{n\geq m} f_{2n-1} \frac{x^{2n-1}}{(2n-1)!}$ for $m\geq 1$. Suppose
$$
F(x)+F(\frac{x}{1+x})=\frac{2[\ln (1+x)]^{2m}}{(2m)!}+\frac{2[\ln(...
2
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1
answer
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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?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...
- 716
2
votes
1
answer
232
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On a lemma of Łojasiewicz in complex analysis of one variable
Context. The question arises from my former question on the remainder of a power series. Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper ...
- 5,288
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votes
1
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66
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Exponential taylor series for multiple variables with linear constraints for coefficients
I'm trying to simplify the sum
$$
\sum_{\vec x \in (\mathbb{N}_0)^n: M\vec x = \vec b} \prod_i \frac{(a_i)^{x_i}}{x_i!},
$$
where $M$ is a $\mathbb{N}_0$-valued $m\times n$ matrix, $\vec b$ is $\...
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Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?
Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...
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Power series expansion of the order parameter in the Kuramoto model
In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
...
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votes
0
answers
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Zeros of inverse of dilogarithm
I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also ...
- 4,461
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Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$
Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series ...
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A few reference questions about the Baker–Campbell–Hausdorff formula
I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.
Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$
$$ \...
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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.
- 21
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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 ...
- 1,826
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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 $...
- 1,087
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answers
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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( \...
- 1,087
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vote
0
answers
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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 ...
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0
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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 $j^{\text{th}}$ (...
- 11.8k
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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 ...
- 499
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1
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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\...
- 63
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1
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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 $...
- 3,561
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2
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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 ...
- 21.2k
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1
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127
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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$ ...
- 5,288
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votes
1
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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 ...
3
votes
0
answers
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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(...
- 5,288
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votes
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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}\...
- 716
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votes
2
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Faster computation of p-adic log
As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting).
When it comes to computing $\log P(x)$, one may use the formula
$$
(\log P)' = \...
- 919
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votes
0
answers
71
views
Which power series in $\mathbb{Z}_p[[T]]$ are rational functions? [duplicate]
Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as:
$$f(...
- 1,799
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1
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What's the fastest way to compute $\log n$ for $n>1$?
As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series
$$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$
Thus, to compute $\log n$ with $n>1$, we may employ the ...
- 13.9k
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votes
0
answers
132
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Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms
Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
1
vote
1
answer
206
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A conjectural identity involving infinite series
Recently I formulated the following curious conjecture based on my computation.
Conjecture. For all $|x|>1$, we have the identity
$$\sum_{k=0}^\infty\frac{\sum_{j=0}^{k}\binom{2k+1}{2j}(1-x)^jx^{k-...
- 13.9k
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votes
0
answers
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Karamata's Abelian/Tauberian Theorem in the complex plane
The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...
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votes
1
answer
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Is the minimal polynomial of an algebraic formal Laurent series always separable?
Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from ...
- 365
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0
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102
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Property of an integer sequence related to series reversion
Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:
Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...
- 15.9k
1
vote
1
answer
294
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Are algebraic power series in positive characteristics D-finite?
We know that in characteristic $0$, all algebraic series are differentiably finite.
Is this true in positive characteristic? I look at the proof, indeed we need to the
characteristic to be $0$ for the ...
- 365
1
vote
2
answers
387
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Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
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