In this message at Stack Exchange, there is an identity involving the Meijer G function for M2 found at A157852 - OEIS.
My question here is how can I code a check to a few upper limits in place of infinity to see if we can make
Of course, the proof is always best, particularly here where the Meijer G function for M2 is very sluggish for large upper limits! (I’ve never gotten over 20.)
For all my work on this form of M2 see
So, how do we code this? the facts are we have
M2=\sum _{n=1}^{\infty } \left(\frac{i}{\pi }\right)^{1-n}\left(\text G_{n+1,n+2}^{n+2,0}\left(^{1;\underbrace {x,…,x}_n}_{1-n,0; \underbrace {x,…,x}_n}\bigg|-\pi i\right)\right).
and this
found here
Tyma Gaidash pointed me to the Wolfram Functions site where I tried the following.
I might be missing more, but I see I’m missing the input for n. How should I add it?
I found a proof for
It was so simple, it looked trivial, but it is indeed omnipotent!
In Mathematica, I entered
f[n_] := MeijerG[{{},Table[1, {n + 1}]}, {Prepend[Table[0, n + 1], -n + 1], {}}, -I \[Pi]];
Integrate[E^(I Pi x) (x^(1/x) - 1), {x, 1, Infinity I}]
and got exactly
Your analysis is still desired.
It turns out that is not proving what I thought, because I entered the wrong line.
Here is some numeric checking, however: