Compute $\sum_{n=1}^\infty\frac{H_nH_{2n}}{(2n+1)^3}$

Question

How to prove that

$$\small{\sum_{n=1}^\infty\frac{H_nH_{2n}}{(2n+1)^3}=\frac1{12}\ln^52+\frac{13}{128}\zeta(5)-\frac12\ln^32\zeta(2)+\frac74\ln^22\zeta(3)-\frac{17}{8}\ln2\zeta(4)+2\ln2\operatorname{Li}_4\left(\frac12\right)}$$ whre $H_n$ is the harmonic number, $\zeta$ is the Riemann zeta function and $\operatorname{Li}_a(x)$ is the polylogarithm function.

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This problem is proposed by Cornel ( can be found here ) and no solution has been submitted yet. I tried all the tools I used in solving the other tough series but did not work, so I consider this series very hard to crack. Any idea ?

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• I am tagging " integration" as logarithmic integrals are very related to harmonic series.

Answer 1

Here is a sketch of Cornel's way.

We use that $\displaystyle \sum_{k=1}^{\infty} \frac{H_k}{(k+1)(k+2n+1)}=\frac{H_{2n}^2+H_{2n}^{(2)}}{4n}$ and then multiply all by $1/n^2$ and consider the summation from $n=1$ to $\infty$. Later in the process we use another critical step, $\displaystyle \sum_{n=1}^{\infty} \frac{1}{(2k+2n+1) 2n}=\frac{1}{(2k+1)^2}+\frac{H_{2k}}{2k+1}-\frac{H_k}{2(2k+1)}-\frac{\log(2)}{2k+1}$.

Essentially, these are almost exactly the steps presented in section 6.59, pages $530$-$532$, in the book (Almost) Impossible Integrals, Sums, and Series.

So, after simple calculations and rearrangements we arrive at

$$\frac{1}{4} \sum _{n=1}^{\infty } \frac{\left(H_{2 n}\right){}^2}{n^3}+\frac{1}{4} \sum _{n=1}^{\infty } \frac{H_{2 n}^{(2)}}{n^3}$$

$$=\frac{1}{8}\sum _{n=1}^{\infty }\frac{H_n}{n^4}-\frac{1}{4}\sum _{n=1}^{\infty }\frac{H_n H_{2 n}}{n^3}+\frac{\pi^2}{24}\sum _{n=1}^{\infty }\frac{H_{2 n}}{n^2}-4\sum _{n=1}^{\infty }\frac{H_{2 n+1}^2}{(2 n+1)^3}+\frac{\pi^2}{6}\sum _{n=1}^{\infty }\frac{H_{2 n+1}}{(2 n+1)^2}\\+4\sum _{n=1}^{\infty }\frac{H_{2 n+1}}{(2 n+1)^4}+2\sum _{n=1}^{\infty }\frac{H_n H_{2 n}}{(2 n+1)^3}+4 \log (2)\sum _{n=1}^{\infty }\frac{ H_{2 n+1}}{(2 n+1)^3}-\frac{\pi^2}{48}\sum _{n=1}^{\infty }\frac{1}{n^3}-\frac{\pi^2}{6}\sum _{n=1}^{\infty }\frac{1}{(2 n+1)^3}-4 \log (2)\sum _{n=1}^{\infty }\frac{1}{(2 n+1)^4}.$$

Since all the series are known except the desired one, the extraction is immediately achieved.

For example, a solution to the challenging series $\displaystyle \sum _{n=1}^{\infty }\frac{H_n H_{2 n}}{n^3}$ is presented in https://math.stackexchange.com/q/3345138.

The way to go also appears in details in the preprint On the calculation of two essential harmonic series with a weight $5$ structure, involving harmonic numbers of the type $H_{2n}$ by Cornel Ioan Valean