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I would like to sum the following $$\sum_{n=1}^{\infty}\frac{1}{n^b(n+m)^b}$$ for values of $b$ for which the sum exists. The problem is that truncating the sum does not work that good, and needs in general large number of terms specially when $b$ is close to non-summability areas. I then tried to rewrite the sum into some function for which we might have convenient and precise ways to calculate them. Unfortunately have not been successful. I appreciate any ideas. Many thanks!

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    $\begingroup$ If $b$ is an integer, you may compute such series through partial fraction decomposition and the functions $\psi,\psi',\psi'',\ldots$ where $\psi(x)=\frac{d}{dx}\log\Gamma(x)$. $\endgroup$ Commented Dec 15, 2016 at 14:48
  • $\begingroup$ That's the problem. $b $ is not an integer. $\endgroup$
    – Math-fun
    Commented Dec 15, 2016 at 14:50
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    $\begingroup$ The apply the inverse Laplace transform to convert your series into an integral. $\endgroup$ Commented Dec 15, 2016 at 14:50
  • $\begingroup$ That sounds interesting I will try. Thank you :-) $\endgroup$
    – Math-fun
    Commented Dec 15, 2016 at 14:51
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    $\begingroup$ In the critical case $b\approx\frac{1}{2}$, you will see that the behaviour of your series depends on the behaviour of a Bessel $I$ function. $\endgroup$ Commented Dec 15, 2016 at 14:52

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