So I have been trying for a few days to figure out the sum of
S=∞∑k=11k2−a2 where a∈(0,1). So far from my nummerical
analysis and CAS that this sum equals
S=12a[1a−πcot(aπ)]
But I have not been able to prove this yet. Anyone know how? My guess is that the
sum of this series is related to fourier-series but nothing particalr comes to mind.
For the easy values, I have been able to use telescopic series, and a bit of algebraic magic, but for the general case I am stumpled. Anyone have any ideas or hints? Cheers.
Answer
This question was settled in the Mathematics chatroom, but I'll put up the solution here for reference.
Starting with the infinite product
sinπxπx=∞∏k=1(1−x2k2)
taking the logarithm of both sides gives
log(sinπxπx)=log(∞∏k=1(1−x2k2))=∞∑k=1log(1−x2k2)
Differentiation gives
πxsinπx(cosπxx−sinπxπx2)=∞∑k=1−2xk2(1−x2k2)
which simplifies to
πcotπx−1x=−2x∞∑k=11k2−x2
or
∞∑k=11k2−x2=12x2−πcotπx2x
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