We can try the harmonic analysis path. Since:
we have, as an example:
hence by multiplying and we can write the Fouries cosine series of over and grab from and grab from a combinatorial equivalent for
With the aid of Mathematica I got:
So we have the Fourier cosine series of but the path does not look promising from here. However, if we replace with a periodic continuation we get the way nicer identity:
that directly leads to:
Now since and , the first two identites are easily proven. Now the three-terms integral
is a linear combination of depending on the parity of , so it is quite difficult to find, explicitly, the Fourier cosine series of , or the integral , but still not impossible. In particular, we know that the Taylor coefficients of the powers of depends on the generalized harmonic numbers. In our case,
hence we can just find a closed form for
and sum everything through the third previous identity. Ugh.
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