I would like to compute sums of the type
\begin{equation} \sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k} \end{equation}
where $P_n(z)$ are Legendre polynomials, $H(n)$ are harmonic numbers and $k = 0, 1, 2, 3...$, a positive integer.
I've tried using the integral representation for $H(n)$ and then the generator function for legendre polynomials. Moreover, I tried using the integral representation for legendre polynomials, but it seems that nothing works...
Any suggestion?