So far, not very promising. E.g., Mathematica does their impressive-looking In[26]:= ContinuedFractionK[-k^6, k^3 + (k + 1)^3, {k, ∞}] Out[26]= -(PolyGamma[2, 2]/PolyGamma[2, 1]) In[27]:= FunctionExpand@% // Expand Out[27]= -1 + 1/Zeta[3] presumably because it's termwise identical to the partial sums of the ζ series: In[22]:= ContinuedFractionK[-k^6, k^3 + (k + 1)^3, {k, 9}] Out[22]= -3161105947/19164113947 In[29]:= Sum[k^-3, {k, 2, 10}] Out[29]= 3161105947/16003008000 In[30]:= 1 + 1/% Out[30]= 19164113947/3161105947 Years ago, Julian & I reexpressed K(quadratic)/(linear) as log-derivatives of pFqs, almost certainly not for the first time. Maybe the project will conjecture some new pFqs. —rwg
Subject: [math-fun] The Ramanujan Machine Date: 2019-07-16 04:34 From: Hans Havermann <gladhobo@bell.net> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>
A story in phys.org yesterday: "A team of researchers at the Israel Institute of Technology has built what they describe as a Ramanujan machine - a device that automatically generates conjectures (mathematical statements that are proposed as true statements) for fundamental constants."
An arXiv preprint is here:
https://arxiv.org/abs/1907.00205
There's also a website that provides overview and welcomes participation:
http://www.ramanujanmachine.com
In spite of the authors being listed in the (above noted) preprint, the website (and a link to its version of the paper) currently still pretends to be "anonymous due to the double-blinded process of the peer-reviewed journal to which we submitted the paper". _______________________________________________