# Questions tagged [power-series]

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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 ...
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### 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 ...
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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 ...
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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 ...
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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. ...
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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-...
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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 ...
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1 vote
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### A problem on monotonicity rule for the ratio of two Maclaurin power series

In the paper  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 ...
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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+\...
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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 ...
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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.
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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 ...
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1 vote
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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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### 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 $j^{\text{th}}$ (...
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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 ...
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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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### 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 ...
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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 ...
1 vote
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 ...
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 \$\...