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Powers of Dilogarithm function

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
This thread will be dedicated to try finding a general solution for the integral


\(\displaystyle \int^x_0 \frac{\operatorname{Li}_2(t)^2}{t}\, dt \,\,\,\,\,\,\, 0\leq x \leq 1\)
We define the following

\(\displaystyle \operatorname{Li}_2(x)^2 =\left( \int^x_0 \frac{\log(1-t)}{t}\, dt \right)^2\)​

This is NOT a tutorial , every member is encouraged to give thoughts on how to solve the question.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Let us try the special case \(\displaystyle x=1\)

\(\displaystyle
\begin{align}

\int^1_0 \frac{\operatorname{Li}_2(x)^2}{x}\, dx &= \int^1_0 \sum_{k\geq 1} \sum_{n\geq 1}\frac{1}{n^2 k^2}\, x^{k+n-1}\, dx \\
& = \sum_{k\geq 1}\sum_{n\geq 1 }\frac{1}{n^2 k^2 (n+k)}\\
& = \sum_{k\geq 1}\frac{1}{k^2} \sum_{n \geq 1}\left( \frac{1}{k \, n^2}-\frac{1}{k^2 \, n}+\frac{1}{k^2(n+k)} \right)\\
& = \sum_{k\geq 1}\frac{1}{k^3} \sum_{n\geq 1}\frac{1}{n^2}-\sum_{k\geq 1}\frac{1}{k^4} \sum_{n \geq 1} \frac{1}{n}-\frac{1}{(n+k)}\\
& = \zeta(3)\zeta(2)- \sum_{k \geq 1}\frac{H_k}{k^4}\\
&= 2\zeta(3)\zeta(2)-3\zeta(5)
\end{align}

\)
 
Last edited:

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
That made me think of finding an analytic solution of the special case \(\displaystyle x=\frac{1}{2}\) . My chance might be slim but I will try it later.
 

Random Variable

Well-known member
MHB Math Helper
Jan 31, 2012
253
The fact that Wolfram Alpha can't express an antiderivative in terms of elementary functions and polylogs (or in terms of any functions for that matter) is a bit discouraging.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
The fact that Wolfram Alpha can't express an antiderivative in terms of elementary functions and polylogs (or in terms of any functions for that matter) is a bit discouraging.
I think the solution if it exists will involve infinite sums of partial sums of poly logarithms. I think we should define a new function.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
The solution of the integral seems to involve the following double sum

\(\displaystyle \sum_{k\geq 1} \sum_{n\geq 1} \frac{x^{n+k}}{(nk)^2 (n+k)}\)

Generally it will be interesting looking at the sum

\(\displaystyle \sum_{a_1\geq 1} \sum_{a_2\geq 1} \cdots \sum_{a_n\geq 1} \frac{x^{a_1+a_2+\cdots +a_n}}{(a_1a_2 \cdots a_n)^k(a_1+a_2+\cdots +a_n)}\)