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Inspired by the comments in Solving $\int_{\mathbb R} \frac{(x^2+a^2)^{1-2s}}{(x^2+1)^{1-s}} dx$ I'm wondering for which $z \in \mathbb C$ the Pfaff Transformation $$_2F_1(a,c-b;c;z/(z-1)) = (1-z)^a {}_2F_1(a,b;c;z)$$ holds. I've got two contradicting sources: source 1 and source 2. By source 1 it would even hold for $z=1$ which I don't understand since the argument $z/(z-1)$ isn't defined there.

principal-ideal-domain
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    Hello again. Here are some notes on the hypergeometric function that you might find helpful. Theorem $4$ provides a proof of one version of the Pfaff transform. https://homepage.tudelft.nl/11r49/documents/wi4006/hyper.pdf – David H Sep 03 '21 at 13:38
  • @DavidH Thanks, that's a nice and easy to read introduction to hypergeometric functions. – principal-ideal-domain Sep 03 '21 at 20:57

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