# 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).

• 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