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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?

Siddhartha Morales
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  • I think in general this isn't going to have a closed form. – K.defaoite Mar 14 '21 at 01:59
  • @K.defaoite I already found a closed form for k=0,1. The problem is that it's quite ugly, so I think that there is a smarter way of computing it. In fact, I have to compute $$ \sum_{n=1}^{\infty} P_{n}(z) \frac{H(n)}{(n+k)^\alpha} $$ for $\alpha$ a positive integer. – Siddhartha Morales Mar 14 '21 at 04:32

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