Let 
denote the 
– th harmonic number of weight 
. Prove that
![\[\sum_{n=1}^{\infty}\frac{\mathcal{H}_n^2- \mathcal{H}_n^{(2)}}{(n+1)(n+2)}=2\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-bd97266705646894b7b056b6581ce534_l3.png "Rendered by QuickLaTeX.com")
Solution
Recall the generating function
(1) 
Thus 
from 
to 
we get that
(2) 
Integrating 
from to 
the result follows.
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Let denote the – th harmonic number of weight . Prove that
Solution
Recall the generating function
(1)
Thus from to we get that
(2)
Integrating from to the result follows.