Challenging Sum: compute $\sum_{n=1}^\infty\frac{H_n}{2n+1}\left(\zeta(3)-H_n^{(3)}\right)$

A second solution in large steps by Cornel Ioan Valean

Let's start with the following useful identity which is easily derived by using recurrence relations and simple rearrangements, manipulations with sums, that is

Let $n$ be a non-negative integer number. Then, we have $$\int_0^1 x^{2n}\frac{\log(1+x)}{1+x}\textrm{d}x$$ $$=\frac{1}{2}H_{2n}^2-2\log(2) H_{2n}+\frac{1}{2}H_{2n}^{(2)}-\frac{1}{4}H_n^2-\frac{1}{4}H_n^{(2)}+\log (2)H_n+\frac{1}{2} \log ^2(2)-\sum_{k=1}^{n-1}\frac{H_k}{2 k+1},$$ where $H_n^{(m)}=1+\frac{1}{2^m}+\cdots+\frac{1}{n^m}$ represents the $n$th generalized harmonic number of order $m$.

By multiplying both sides of the identity above by $1/n^3$ and considering the summation from $n=1$ to $\infty$, we get

$$\sum_{n=1}^{\infty} \frac{1}{n^3}\sum_{k=1}^{n-1}\frac{H_{k}}{2 k+1}=\sum_{k=1}^{\infty} \sum_{n=k+1}^{\infty}\frac{1}{n^3}\frac{H_{k}}{2 k+1}=\underbrace{\sum_{k=1}^{\infty}\frac{H_{k}}{2 k+1}\left(\zeta(3)-H_k^{(3)}\right)}_{\text{The desired series}}$$ $$=\frac{1}{2}\sum_{n=1}^{\infty}\frac{H_{2n}^2}{n^3}-2\log(2) \sum_{n=1}^{\infty}\frac{H_{2n}}{n^3}+\frac{1}{2}\sum_{n=1}^{\infty}\frac{H_{2n}^{(2)}}{n^3}-\frac{1}{4}\sum_{n=1}^{\infty}\frac{H_n^2}{n^3}-\frac{1}{4}\sum_{n=1}^{\infty} \frac{H_n^{(2)}}{n^3}$$ $$+\log (2)\sum_{n=1}^{\infty} \frac{H_n}{n^3}+\frac{1}{2}\log ^2(2)\sum_{n=1}^{\infty}\frac{1}{n^3}-\int_0^1 \frac{\log(1+x)}{1+x}\operatorname{Li}_3(x^2)\textrm{d}x,$$

where we see all the series in the right-hand side are easily reducible to known series which may also be found in the book (Almost) Impossible Integrals, Sums, and Series.

On the other hand, with simple integration by parts, we obtain $$\int_0^1 \frac{\log(1+x)}{1+x}\operatorname{Li}_3(x^2)\textrm{d}x$$ $$=\frac{1}{2}\log^2(2)\zeta(3)-2\int_0^1 \frac{\log^2(1+x)\operatorname{Li}_2(x)}{x}\textrm{d}x-2\int_0^1 \frac{\log^2(1+x)\operatorname{Li}_2(-x)}{x}\textrm{d}x,$$ where the last integrals may be found calculated in the paper The calculation of a harmonic series with a weight $5$ structure, involving the product of harmonic numbers, $H_n H_{2n}^{(2)}$.

A note: The sister of the result above (easy to obtain by recurrence relations and very useful),

$$\int_0^1 x^{2n-1} \frac{\log(1+x)}{1+x}\textrm{d}x$$ $$=2\log(2) H_{2n}-\log(2)H_n+\frac{1}{4}H_n^2+\frac{1}{4}H_n^{(2)}-\frac{1}{2}H_{2n}^2-\frac{1}{2} H_{2n}^{(2)}+\frac{H_{2n}}{2n}-\frac{H_n}{2n} $$ $$ -\frac{1}{2}\log^2(2)+\sum_{k=1}^{n-1}\frac{H_k}{2 k+1}. $$

A first solution in large steps by Cornel Ioan Valean

Using The Master Theorem of Series in the article A master theorem of series and an evaluation of a cubic harmonic series or from the book, (Almost) Impossible Integrals, Sums, and Series, pages $288$-$289$, where we set $\mathcal{M}(k)=H_{2k}$, $\displaystyle m(k)=H_{2k}-H_{2k-2}=\frac{1}{2k}+\frac{1}{2k-1}$, we get

$$\sum_{k=1}^{\infty}\frac{H_{2k}}{(k+1)(k+n+1)}$$ $$=\frac{1}{4}\frac{H_n^2}{n}-\log(2)\frac{H_n}{n}+2\log(2)\frac{H_{2n}}{n}+\frac{1}{4}\frac{H_n^{(2)}}{n}-\frac{4\log(2)}{2n+1}+\frac{1}{n}\sum_{k=1}^n \frac{H_k}{2k+1}.$$

Multiplying both sides of the result above by $1/n^2$ and considering the sum from $n=1$ to $\infty$, we have $$\frac{5}{2}\zeta(4)-\frac{1}{2}\zeta(2)\zeta (3)-4\log (2)\zeta(2)+4\sum _{n=1}^{\infty } \frac{H_n}{n(2n-1)}-2\sum _{n=1}^{\infty } \frac{H_n}{n^2}-\sum _{n=1}^{\infty } \frac{H_n}{n^3}$$ $$+\frac{1}{2}\sum _{n=1}^{\infty } \frac{H_n}{n^4}+\zeta(2)\sum _{n=1}^{\infty } \frac{H_{2 n}}{n^2}-\underbrace{\sum _{n=1}^{\infty } \frac{H_n H_{2 n}}{n^3}}_{\text{A tough series}}$$ $$=16 \log (2)-16 \log ^2(2)-4\log (2)\zeta(2)+\frac{1}{4}\sum _{n=1}^{\infty } \frac{H_n^2}{n^3}+\sum _{n=1}^{\infty } \frac{H_n}{(2 n+1) n^3}-\log (2)\sum _{n=1}^{\infty } \frac{ H_n}{n^3}$$ $$+2 \log (2)\sum _{n=1}^{\infty } \frac{ H_{2 n}}{n^3}+\frac{1}{4}\sum _{n=1}^{\infty } \frac{H_n^{(2)}}{n^3}+\underbrace{\sum _{n=1}^{\infty } \frac{H_n}{2 n+1}\left(\zeta (3)-H_n^{(3)}\right)}_{\text{The desired series}},$$ where in the calculations we used the folowing result from the paper On the calculation of two essential harmonic series with a weight $5$ structure, involving harmonic numbers of the type $H_{2n}$, $$\sum _{n=1}^{\infty } \frac{H_n H_{2 n}}{(2 n)^3}$$ $$=\frac{307}{128}\zeta(5)-\frac{1}{16}\zeta (2) \zeta (3)+\frac{1}{3}\log ^3(2)\zeta (2) -\frac{7}{8} \log ^2(2)\zeta (3)-\frac{1}{15} \log ^5(2)$$ $$-2 \log (2) \operatorname{Li}_4\left(\frac{1}{2}\right) -2 \operatorname{Li}_5\left(\frac{1}{2}\right),$$ and at the same time we used that by reversing the summation order our desired series is (almost) revealed $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^3}\sum_{k=1}^n \frac{H_k}{2k+1}=\sum _{n=1}^{\infty } \frac{H_n}{2 n+1}\left(\zeta (3)-H_n^{(3)}+\frac{1}{n^3}\right).$$

The rest of the series are well-known (or immediately reducible to well-known series), and all of them may also be found calculated in the book, (Almost) Impossible Integrals, Sums, and Series.

A note: By using the same procedure, we may calculate the series $$\sum _{n=1}^{\infty } \frac{H_n}{2 n+1}\left(\zeta (2)-H_n^{(2)}\right),$$ or other versions if we know the resulting series after applying this strategy.