Re: An exact simplification challenge - 37 (MeijerG) - Go and surpass all CASs

Hi Vladimir,

The answer is 12*Sqrt[2]*Pi, but unlike your previous challenges, I
did
not particularly like this one. It seems too artificial. Could you
share
your path to creating this challenge ?

But, anyway, here is how you arrive to the answer:

Start with the following function of z:

In[1]:= res =
MeijerG[{{1/12, 1/4, 5/12, 7/12, 3/4, 11/12}, {}}, {{-(1/6), 1/3, 1/
2, 5/6, 7/6}, {-(1/3), 0, 1/6, 1/3, 1/2, 2/3, 1}}, z] +
MeijerG[{{1/12, 1/4, 5/12, 7/12, 3/4, 5/6, 11/12}, {}}, {{1/3, 1/2,
5/6, 5/6, 7/6}, {-(1/3), -(1/6), 0, 1/6, 1/3, 1/2, 2/3, 1}},
z];

(* series expansion around the origin to different degrees *)

In[2]:= Normal[
Table[FullSimplify[Series[%, {z, 0, k}]], {k, 5, 20, 5}]]

Out[2]= {(4 Sqrt[2] \[Pi])/z^(1/6), (4 Sqrt[2] \[Pi])/z^(1/6), (
4 Sqrt[2] \[Pi])/z^(1/6), (4 Sqrt[2] \[Pi])/z^(1/6)}

In[3]:= First[%] /. z -> 1/729

Out[3]= 12 Sqrt[2] \[Pi]

In[4]:= N[res /. z -> 1/729, 150]/Pi // RootApproximant

Out[4]= 12 Sqrt[2]

Funny enough, the Plouffe's inverter fails to recognize this numerical
value
as the derived above constant.

Post-factum one proves that res is a simple function as follows:

In[5]:= res2 = Expand[z^(1/6) res, _MeijerG]

Out[5]= MeijerG[{{1/4, 5/12, 7/12, 3/4, 11/12, 13/12}, {}}, {{0, 1/2,
2/3, 1, 4/3}, {-(1/6), 1/6, 1/3, 1/2, 2/3, 5/6, 7/6}}, z] +
MeijerG[{{1/4, 5/12, 7/12, 3/4, 11/12, 1, 13/12}, {}}, {{1/2, 2/3, 1,
1, 4/3}, {-(1/6), 0, 1/6, 1/3, 1/2, 2/3, 5/6, 7/6}}, z]

In[6]:= D[res2, z]

Out[6]= 0


Oleksandr Pavlyk
Wolfram Research Inc

On Jul 29, 2:19 am, Vladimir Bondarenko <v...@xxxxxxxxxxxxxxx> wrote:
Hello symbolic calculation fans,

None of the modern computer algebra systems is able to crack
this straightforwardly.

Is there a Whiz the Simplifier to come up with the steps to
compress (very much :) this sum

MeijerG[{{1/12,1/4,5/12,7/12,3/4,11/12}, {}},
{{-1/6,1/3,1/2,5/6,7/6}, {-1/3,0,1/6,1/3,1/2,2/3,1}}, 1/729]

+

MeijerG[{{1/12,1/4,5/12,7/12,3/4,5/6,11/12}, {}},
{{1/3,1/2,5/6,5/6,7/6}, {-1/3,-1/6,0,1/6,1/3,1/2,2/3,1}}, 1/729]

?

Best wishes,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLChttp://maple.bug-list.org/ Maple Bugs Encyclopaediahttp://www.CAS-testing.org/ CAS Testing


.