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Tagged with polylogarithm generating-functions
6
questions
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Asymptotic expansions of sums via Cauchy's integral formula and Watson's lemma
I am looking for references that deal with the asymptotic expansions of sums of the form
$$s(n)=\sum_{k=0}^n g(n,k)$$
using the (or similar to) following method.
We have the generating function
$$f(z)=...
- 2,249
12
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2
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How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
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Generating function of Clausen functions: $\displaystyle\sum_{n=1}^\infty \text{Cl}_{2n}(x)\frac{t^{2n}}{(2n)!}$
Context
I was trying to solve a series:
$$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$
Using the Fourier series
\begin{equation}
\begin{split}
\log\Gamma(x)&=\frac{\ln(2\pi)}{2}+\sum_{k=1}^\infty \frac{\...
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Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function
$\def\W{\operatorname W} \def\Li{\operatorname{Li}} $
Interested by $\sum_\limits{n=1}^\infty\frac{\W(n^2)}{n^2}$, here is an example where Lagrange reversion applies to a Lambert W sum:
$$\W(x)=\ln(...
- 12.5k
3
votes
1
answer
502
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Generating function of the polylogarithm.
Let $\operatorname{Li}_s(z)$ denote the polylogarithm function
$$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}.$$
Does there exists a closed form or a known function which generates the ...
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A group of important generating functions involving harmonic number.
How to prove the following identities:
$$\small{\sum_{n=1}^\infty\frac{H_{n}}{n^2}x^{n}=\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)+\ln(1-x)\operatorname{Li}_2(1-x)+\frac12\ln x\ln^2(1-x)+\zeta(3)...
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