# Expanding derivatives of hypergeometric functions

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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yes, the HypExp package does exactly that. Includes also some hypergeometric simplifications not found in Mathematica for some reason. –  lalmei Jul 23 '14 at 23:54
@lalmei Why don't you provide an answer with a link to that package and a few examples? –  Mr.Wizard Jul 24 '14 at 8:09
@Mr.Wizard Thanks for the suggestion, I will provide some examples soon. –  lalmei Jul 24 '14 at 13:18

## 1 Answer

The Package HypExp does exactly that. Here is the link to 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 has been some work on the simplification of polylogarithms into a Hopt Algebras, which simplifies the reduction of the Hypergeometic 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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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? –  Mr.Wizard Jul 24 '14 at 13:21
@Mr.Wizard Wow, I thought it was a free repository, but apparently it is only for "institutional members." The Arxiv paper states "Lincense: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. –  lalmei Jul 24 '14 at 14:00
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.) –  Mr.Wizard Jul 24 '14 at 14:06