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Need more help on code for identity involving the Meijer G .

In this message at Stack Exchange, there is an identity involving the Meijer G function for M2 found at https://oeis.org/A157852.

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

enter image description here 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

POSTED BY: Marvin Ray Burns
4 Replies

It turns out that is not proving what I thought, because I entered the wrong line.

Here is some numeric checking, however:

POSTED BY: Marvin Ray Burns

I found a proof for $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). $

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 $$\int_1^{i \infty } e^{i \pi x} \left(x^{1/x}-1\right) \, dx.$$

Your analysis is still desired.

POSTED BY: Marvin Ray Burns

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?

POSTED BY: Marvin Ray Burns

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

enter image description here

found here

POSTED BY: Marvin Ray Burns
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