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Browsing this site and the web I could not find a reference on the following combinations of Gauss hypergeometric functions $F={}_2F_1$, for which I have reason to believe that they can be simplified or even expressed in terms of elementary functions:

For $x>0$ and $z\in \mathbb C$ $$ xF(2,z,z+1;-x)+x^{-1}F(2,z,z+1;-x^{-1})\quad=\quad ? $$ $$ F(1,z,z+1;x)+F(1,z,z+1;x^{-1})\quad=\quad ? $$ Any comments are appreciated.

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  • $\begingroup$ "I have reason to believe that they can be simplified" ... What is the reason? $\endgroup$
    – Iosif Pinelis
    Commented Sep 26, 2023 at 13:09
  • $\begingroup$ @IosifPinelis: They come up in the calculation of the resolvent kernel of the d'Alembertian on 3-dimensional anti-de Sitter space. In the physics literature there are some formulas to which I can compare my calculations and these formulas do not involve hypergeometric functions. I hope that by simplifying the above expressions I get something that I can compare. $\endgroup$
    – B K
    Commented Sep 26, 2023 at 13:23
  • $\begingroup$ Mathematica cannot do anything with these expressions. So, any simplification of them seems unlikely. $\endgroup$
    – Iosif Pinelis
    Commented Sep 27, 2023 at 2:19

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