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calculus and analysis - 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).



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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