I'm looking for a reference and/or table for double summations. The sum I'm trying to compute is
$$\sum_{k=1}^\infty \sum_{m=1}^\infty \frac{1}{km(ak^2+bm^2)}$$ for real numbers $a$, $b$.
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$\begingroup$ Closed form formula is, probably, out of question. What do you really want to know about that sum? $\endgroup$– fedjaCommented May 26, 2018 at 23:08
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$\begingroup$ Can you do the special case $a=b=1$ in closed form? If not, then the general case is (of course) hopeless. $\endgroup$– Gerald EdgarCommented Jun 25, 2018 at 11:42
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1 Answer
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You need not any tables to this end. Mathematica does the job by
f[a_, b_] := NSum[1/(k*n*(a*k^2 + b*n^2)), {k, 1, Infinity}, {n, 1, Infinity}]
For example,
f[Pi,0.1]
$0.944319 $
I have strong doubts concerning a closed-form expression for this sum.
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4$\begingroup$ I think OP is looking for closed-form expressions. Evaluating them numerically is easy to do in any programming language. $\endgroup$ Commented May 26, 2018 at 10:14
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$\begingroup$ @Federico Poloni: Can you present something constructive? Both Maple and Mathematica fail with it. $\endgroup$ Commented May 26, 2018 at 10:17
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4$\begingroup$ No --- I can't answer the question, I'm just pointing out that yours isn't (probably) the answer OP is looking for, and that "you need not any tables" is a very misleading statement. $\endgroup$ Commented May 26, 2018 at 10:24