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Tagged with polylogarithm stirling-numbers
4
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Polylogarithms of negative integer order
The polylogarithms of order $s$ are defined by
$$\mathrm{Li}_s (z) = \sum_{k \geqslant 1} \frac{z^k}{k^s},
\quad |z| < 1.$$
From the above definition, derivatives for the polylogarithms ...
- 11.8k
3
votes
1
answer
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Find the series expansion of $\frac{\ln^4(1-x)}{1-x}$
How to prove that
$$\frac{\ln^4(1-x)}{1-x}=\sum_{n=1}^\infty\left(H_n^4-6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2-6H_n^{(4)}\right)x^n=S_n$$
where $H_n^{(a)}=\sum_{k=1}^n\frac1{k^a}$...
- 25.8k
1
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1
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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
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vote
0
answers
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Polylogarithm further generalized
Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
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