Questions tagged [lacunary-series]

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23 votes
1 answer
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Can we just use the linear term of exponential sums to sum divergent series

Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $ You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
Sidharth Ghoshal's user avatar
2 votes
1 answer
107 views

Uniformization of Julia sets and lacunary series

If the complement of a Julia set of quadratic polynomial z^2+c is locally connected and simply connected, it is uniformized by the complement of the unit disk. Consider the uniformization map and its ...
0x11111's user avatar
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6 votes
0 answers
166 views

Computing residues at $\infty$

As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
Caleb Briggs's user avatar
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8 votes
1 answer
486 views

Is anything known about the power series $\sum x^p$ for $p$ prime?

I'm interested in information about the power series $$\sum_{\text{$p$ prime}} x^p$$ and the related power series $$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$ where $p(n)$ is the nth prime. Immediately, the ...
Caleb Briggs's user avatar
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4 votes
1 answer
377 views

How to correctly renormalize this function at the pole $x=1$? Evaluating: $\sum_{n=1}^{\infty} e^{e^n}$

So I was considering the divergent everywhere but 0 power series $$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$ Now one can do the following "questionable" manipulation $$ f(x) = \sum_{n=0}^{\...
Sidharth Ghoshal's user avatar
1 vote
0 answers
67 views

Mandelbrot's lacunarity realized by fractal or stochastic field?

It is my understanding that Mandelbrot came up with the notion of lacunarity to classify the homogeneity of 2D functions that only take two distinct values see here. I wonder, does there exist a ...
Guido Li's user avatar
4 votes
1 answer
493 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, ...
Szeto's user avatar
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11 votes
1 answer
750 views

Distribution of zeroes of lacunary functions

In a recent Math Stack Exchange question I asked about the function $$f(z)=\sum_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary ...
Semiclassical's user avatar
1 vote
1 answer
977 views

Lacunary sequence

Is there a standard definition for a lacunary sequence? Suppose $0 < a_1 < a_2 < \cdots.$ I've read two papers using the term recently. One requires $$ \liminf_n\frac{a_{n+1}}{a_n}>1 $$ ...
Charles's user avatar
  • 8,954
2 votes
1 answer
193 views

Coefficients of lacunary series on quasiconformally transformed unit disk

Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $|q|...
graveolensa's user avatar