given the Lambert series
$$ f(q)= \sum_{n=1}^{\infty}n^{r}\frac{q^{n}}{1-q^{n}} $$
for an real number 'r'
is there a closed form or a soution of this in terms of a differential or integral equation ?
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see the second sum which is not simple at all – Simply Beautiful Art Sep 28 '16 at 17:37
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If $r$ is a positive odd integer or $r = -1$ then it has a closed form in terms of elliptic integral $K$ and modulus $k$. I don't know what happens when $r$ takes other values. Such sums were studied in detail by Ramanujan in his classic paper On certain arithmetical functions. see http://math.stackexchange.com/a/1944103/72031 for odd positive $r$ and for $r =-1$ see http://math.stackexchange.com/a/938644/72031 – Paramanand Singh Sep 30 '16 at 10:41