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Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify these derivatives using FunctionExpand or FullSimplify. In some cases I was able to express those derivatives in terms of elementary functions and well-known mathematical constants, but it required some manual work and was on case-by-case basis. Now I have a table of about a hundred of derivatives I already dealt with and a function that can automatically replace them by their values. For example, it contains cases like

Derivative[0, 1, 0, 0][Hypergeometric2F1][-1/2, 3/2, 1/2, 1/Sqrt[2]] == 
    (Sqrt[1 + Sqrt[2]] (Sqrt[2] Log[3/2 - Sqrt[2]] + 2 (Sqrt[2] + Log[2 + Sqrt[2]])) 
        - 4 Sqrt[2] ArcTan[Sqrt[1 + Sqrt[2]]])/2^(3/4)

and

Derivative[2, 0, 0, 0][Hypergeometric2F1][0, -3/4, 1, 1] == 
    4 π/3 + 13 π^2/12 - 8 Log[2] - 3 π Log[2] + 9 Log[2]^2 - 8 Catalan

I wonder if anybody else tried to solve this problem and found a more general or automated approach to this? Or if anybody has a more comprehensive table of derivatives and is willing to share it?

I can publish my table if anybody is interested (but I haven't kept all calculations that yielded those results).

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  • $\begingroup$ yes, the HypExp package does exactly that. Includes also some hypergeometric simplifications not found in Mathematica for some reason. $\endgroup$
    – lalmei
    Commented Jul 23, 2014 at 23:54
  • $\begingroup$ @lalmei Why don't you provide an answer with a link to that package and a few examples? $\endgroup$
    – Mr.Wizard
    Commented Jul 24, 2014 at 8:09
  • $\begingroup$ @Mr.Wizard Thanks for the suggestion, I will provide some examples soon. $\endgroup$
    – lalmei
    Commented Jul 24, 2014 at 13:18

1 Answer 1

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The Package HypExp does exactly that. Here is the link to the paper for what I believe was the last extension.

After digging around a bit, the package files should be available here ( Edit freely available link)

Several years ago, there was some work on the simplification of polylogarithms into a Hopt Algebras, which simplifies the reduction of the Hypergeometric functions in a much faster and simpler way, this would only be useful if you have pages of Hypergeometric functions you need to simplify. I believe there is a package for that too, I used it once, but that was several laptops ago, and don't remember its name if I find it I will post here.

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  • $\begingroup$ Sadly: "Your IP address is not registered with CPC." Would you please check to see if this package may be freely distributed, and if so upload it elsewhere? $\endgroup$
    – Mr.Wizard
    Commented Jul 24, 2014 at 13:21
  • $\begingroup$ @Mr.Wizard Wow, I thought it was a free repository, but apparently it is only for "institutional members." The Arxiv paper states "License:None." The Final published one has the CPC license, which requires the authors authorization. I sent one of the authors an email let's see how it goes. $\endgroup$
    – lalmei
    Commented Jul 24, 2014 at 14:00
  • $\begingroup$ Thanks. I look forward to their reply. I do not begrudge the authors a fee but in my recollection they see none of it; it's all for the repository. (I hope I'm wrong.) $\endgroup$
    – Mr.Wizard
    Commented Jul 24, 2014 at 14:06
  • $\begingroup$ hypexp.hepforge.org $\endgroup$
    – BabaYaga
    Commented Oct 4, 2023 at 15:49

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