All Questions
10
questions
3
votes
3
answers
388
views
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 862
7
votes
2
answers
132
views
Series expansion of $\text{Li}_3(1-x)$ at $x \sim 0$
My question is simple, but maybe hard to answer. I would like to have a series expansion for $\text{Li}_3 (1-x)$ at $x \sim 0$ in the following form:
$$\text{Li}_3 (1-x) = \sum_{n=0} c_n x^n + \log x \...
- 697
5
votes
2
answers
368
views
Closed form of the sum $s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}$
I am interested to know if the following sum has a closed form
$$s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}\tag{1}$$
I stumbled on this question while studying a very useful book about ...
- 12.7k
4
votes
1
answer
82
views
Evaluate the following series sum.
Problem
I’m trying to evaluate the following series sum
\begin{equation}
S_{j}(z) = \sum_{k=1}^{\infty} \frac{2 H_{k} z^{k+2}}{(k+1)(k+2)^{j}}
\end{equation}
Where
\begin{equation}
H_{k} = \sum_{n=1}^{...
- 87
1
vote
1
answer
59
views
Further Stirling number series resummation
\begin{equation}
\sum_{m=1}^\infty\sum_{n=1}^\infty (-1)^{n } \frac{S_m^{(3)}}{m! n}(-1 + u)^{(m + n - 1)} (\frac{x}{-1 + x})^m
\end{equation}
Note: $S^{(3)}_m$ belongs to the Stirling number of the ...
- 99
0
votes
0
answers
59
views
Stirling number series resummation
\begin{equation}\sum_{m=1}^{\infty}\frac{a_1^3 S_m^{(3)} (u-1)^{m-1}
\left(\frac{x}{x-1}\right)^m}{m!}\end{equation}
Does somebody know the result of this resummation?
Note:
$S_m^{(3)} $ belongs to ...
- 99
0
votes
0
answers
82
views
General expression of a triangle sequence
\begin{gather*}
\frac{1}{4} \\
\frac{1}{4} \quad \frac{1}{4} \\
\frac{11}{48} \quad \frac{1}{4} \quad \frac{11}{48} \\
\frac{5}{24} \quad \frac{11}{48} \quad \frac{11}{48} \quad \frac{5}{24} \\
\frac{...
- 99
0
votes
1
answer
53
views
General expression of a (maybe 3 or 2 dim) sequence [closed]
$\frac{1}{2}$
$\frac{1}{4}$ $\frac{1}{2}$
$\frac{1}{6}$ $\frac{1}{4}$ $\frac{11}{24}$
$\frac{1}{8}$ $\frac{1}{6}$ $\frac{11}{48}$ $\frac{5}{12}$
$\frac{1}{10}$ $\frac{1}{8}$ $\frac{11}{...
- 99
2
votes
0
answers
74
views
How can I prove that the following function is increasing in $x \in [0,1]$?
How can I prove that the following function is increasing in $x$: $$\sum_{i=1}^{\infty} x (1-x) ^ {i-1} \log \left (1+ \mu (1-x)^{i-1} \right)$$
where $\mu$ is any non-negative number and $x$ is in $[...
- 21
16
votes
1
answer
551
views
Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points
The polylogarithm is defined by the series
$$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$
There are relations connecting values of the polylogarithm at certain rational points in the ...
- 47.3k