(1)

\begin{align*} \sum_{k=1}^{n}H_{k} & =\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{j}\\ & =\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{j}\\ & =\sum_{j=1}^{n}\frac{n+1-j}{j}\\ & =\sum_{j=1}^{n}\left(\frac{n+1}{j}-1\right)\\ & =\left(n+1\right)H_{n}-n \end{align*}

(2)

\begin{align*} H_{n,0} & =\sum_{k=1}^{n}\frac{1}{k^{0}}\\ & =\sum_{k=1}^{n}1\\ & =n \end{align*}

(3)

\begin{align*} \sum_{k=1}^{n}H_{k,m} & =\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{j^{m}}\\ & =\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{j^{m}}\\ & =\sum_{j=1}^{n}\frac{n+1-j}{j^{m}}\\ & =\sum_{j=1}^{n}\left(\frac{n+1}{j^{m}}-\frac{1}{j^{m-1}}\right)\\ & =\left(n+1\right)H_{n,m}-H_{n,m-1} \end{align*}

(4)

\begin{align*} \sum_{k=1}^{n}\frac{H_{k}}{k} & =\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{kj}\\ & =\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{kj}\\ & =\sum_{j=1}^{n}\frac{H_{n}-H_{j-1}}{j}\\ & =H_{n}^{\;2}-\sum_{j=1}^{n}\frac{H_{j}-\frac{1}{j}}{j}\\ & =H_{n}^{2}-\LHS+\sum_{j=1}^{n}\frac{1}{j^{2}}\\ & =\frac{1}{2}\left(H_{n}^{\;2}+H_{n,2}\right) \end{align*}

(5)

\begin{align*} \sum_{k=1}^{n}\frac{H_{k,m}}{k^{l}} & =\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{k^{l}j^{m}}\\ & =\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{k^{l}j^{m}}\\ & =\sum_{j=1}^{n}\frac{H_{n,l}-H_{j-1,l}}{j^{m}}\\ & =\sum_{j=1}^{n}\frac{1}{j^{m}}\left(H_{n,l}-H_{j,l}+\frac{1}{j^{l}}\right)\\ & =H_{n,l}\sum_{j=1}^{n}\frac{1}{j^{m}}-\sum_{j=1}^{n}\frac{H_{j,l}}{j^{m}}+\sum_{j=1}^{n}\frac{1}{j^{l+m}}\\ & =H_{n,l}H_{n,m}-\sum_{j=1}^{n}\frac{H_{j.l}}{j^{m}}+H_{n,l+m}\\ & =H_{n,l}H_{n,m}+H_{n,l+m}-\sum_{j=1}^{n}\frac{H_{j.l}}{j^{m}} \end{align*}

(6)

\begin{align*} \sum_{k=1}^{n}\frac{H_{k,m}}{k^{m}} & =H_{n,m}H_{n,m}-\sum_{j=1}^{n}\frac{H_{j.m}}{j^{m}}-\sum_{j=1}^{n}H_{n,2m}\\ & =H_{n,m}^{\;\;\;2}-\LHS-\sum_{j=1}^{n}H_{n,2m}\\ & =\frac{1}{2}\left(H_{n,m}^{2}-\sum_{j=1}^{n}H_{n,2m}\right) \end{align*}

(7)

\begin{align*} \sum_{k=1}^{n}H_{k}^{\;2} & =\sum_{k=1}^{n}\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{1}{ij}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\sum_{j=1}^{k}\frac{1}{ij}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\frac{H_{k}}{i}\\ & =\sum_{i=1}^{n}\frac{1}{i}\left\{ \left(n+1\right)H_{n}-n-\left(iH_{i-1}-\left(i-1\right)\right)\right\} \\ & =\sum_{i=1}^{n}\frac{1}{i}\left\{ \left(n+1\right)H_{n}-n-\left(i\left(H_{i}-\frac{1}{i}\right)-\left(i-1\right)\right)\right\} \\ & =\sum_{i=1}^{n}\frac{1}{i}\left\{ \left(n+1\right)H_{n}-n-\left(iH_{i}-i\right)\right\} \\ & =\left(\left(n+1\right)H_{n}-n\right)\sum_{i=1}^{n}\frac{1}{i}-\sum_{i=1}^{n}\left(H_{i}-1\right)\\ & =\left(\left(n+1\right)H_{n}-n\right)H_{n}-\left(\left(n+1\right)H_{n}-n-n\right)\\ & =\left(n+1\right)H_{n}^{\;2}-\left(2n+1\right)H_{n}+2n \end{align*}

(8)

\begin{align*} \sum_{k=1}^{n}H_{k,l}H_{k,m} & =\sum_{k=1}^{n}\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{1}{i^{l}j^{m}}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\sum_{j=1}^{k}\frac{1}{i^{l}j^{m}}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\frac{H_{k,m}}{i^{l}}\\ & =\sum_{i=1}^{n}\frac{1}{i^{l}}\left\{ \left(n+1\right)H_{n,m}-H_{n,m-1}-\left(iH_{i-1,m}-H_{i-1,m-1}\right)\right\} \\ & =\sum_{i=1}^{n}\frac{1}{i^{l}}\left\{ \left(n+1\right)H_{n,m}-H_{n,m-1}-\left(i\left(H_{i,m}-\frac{1}{i^{m}}\right)-\left(H_{i,m-1}-\frac{1}{i^{m-1}}\right)\right)\right\} \\ & =\sum_{i=1}^{n}\frac{1}{i^{l}}\left\{ \left(n+1\right)H_{n,m}-H_{n,m-1}-\left(iH_{i,m}-H_{i,m-1}\right)\right\} \\ & =\left(\left(n+1\right)H_{n,m}-H_{n,m-1}\right)\sum_{i=1}^{n}\frac{1}{i^{l}}-\sum_{i=1}^{n}\left(\frac{H_{i,m}}{i^{l-1}}-\frac{H_{i,m-1}}{i^{l}}\right)\\ & =\left(\left(n+1\right)H_{n,m}-H_{n,m-1}\right)H_{n,l}-\sum_{i=1}^{n}\left(\frac{H_{i,m}}{i^{l-1}}-\frac{H_{i,m-1}}{i^{l}}\right) \end{align*}

(9)

\begin{align*} \sum_{k=1}^{n}\frac{H_{k}^{2}}{k} & =\sum_{k=1}^{n}\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{1}{kij}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\sum_{j=1}^{k}\frac{1}{kij}\\ & =\sum_{i=1}^{n}\sum_{k=i}^{n}\frac{H_{k}}{ki}\\ & =\frac{1}{2}\sum_{i=1}^{n}\frac{1}{i}\left\{ H_{n}^{2}+H_{n,2}-\left(H_{i-1}^{2}+H_{i-1,2}\right)\right\} \\ & =\frac{1}{2}\sum_{i=1}^{n}\frac{1}{i}\left\{ H_{n}^{2}+H_{n,2}-\left(\left(H_{i}-\frac{1}{i}\right)^{2}+H_{i,2}-\frac{1}{i^{2}}\right)\right\} \\ & =\frac{1}{2}\sum_{i=1}^{n}\frac{1}{i}\left\{ H_{n}^{2}+H_{n,2}-\left(H_{i}^{2}-2\frac{H_{i}}{i}+H_{i,2}\right)\right\} \\ & =\frac{1}{2}\left\{ \left(H_{n}^{2}+H_{n,2}\right)H_{n}-\LHS+2\sum_{i=1}^{n}\frac{H_{i}}{i^{2}}-\sum_{i=1}^{n}\frac{H_{i,2}}{i}\right\} \\ & =\frac{1}{3}\left\{ \left(H_{n}^{2}+H_{n,2}\right)H_{n}+2\sum_{i=1}^{n}\frac{H_{i}}{i^{2}}-\sum_{i=1}^{n}\frac{H_{i,2}}{i}\right\} \\ & =\frac{1}{3}\left\{ \left(H_{n}^{2}+H_{n,2}\right)H_{n}+2\sum_{i=1}^{n}\frac{H_{i}}{i^{2}}-\left(H_{n}H_{n,2}+H_{n,3}-\sum_{j=1}^{n}\frac{H_{j}}{j^{2}}\right)\right\} \cmt{\because\sum_{i=1}^{n}\frac{H_{i,2}}{i}=H_{n}H_{n,2}+H_{n,3}-\sum_{j=1}^{n}\frac{H_{j}}{j^{2}}}\\ & =\frac{1}{3}\left\{ H_{n}^{3}-H_{n,3}+3\sum_{i=1}^{n}\frac{H_{i}}{i^{2}}\right\} \end{align*}

(10)

\begin{align*} H_{n,m} & =\sum_{k=1}^{n}\frac{1}{k^{m-1}k}\\ & =\sum_{k=1}^{n}\left(H_{k,m-1}-H_{k-1,m-1}\right)\frac{1}{k}\\ & =\sum_{k=1}^{n}\frac{H_{k,m-1}}{k}-\sum_{k=1}^{n}\frac{H_{k-1,m-1}}{k}\\ & =\sum_{k=1}^{n}\frac{H_{k,m-1}}{k}-\sum_{k=0}^{n-1}\frac{H_{k,m-1}}{k+1}\\ & =\sum_{k=1}^{n-1}\frac{H_{k,m-1}}{k}-\sum_{k=1}^{n-1}\frac{H_{k,m-1}}{k+1}+\frac{H_{n,m-1}}{n}\\ & =\sum_{k=1}^{n-1}H_{k,m-1}\left(\frac{1}{k}-\frac{1}{k+1}\right)+\frac{H_{n,m-1}}{n}\\ & =\sum_{k=1}^{n-1}\frac{H_{k,m-1}}{k\left(k+1\right)}+\frac{H_{n,m-1}}{n} \end{align*}