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Is there some approximation or regularization that goes in tacitly in the following equality:

$Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}\right) z^{\lambda +1}} \; {}_2F_1\left(\frac{\lambda +1}{2},\frac{\lambda +2}{2};\lambda +\frac{3}{2};\frac{1}{z^2}\right)$

I am writing the above equation as a particular case of the identity given here. In particular if I do an expansion of both the functions around $z=\infty$ in Mathematica, the function $Q_{\lambda }(z)$ has some imaginary part that doesn't seem to be captured by RHS, which has a nice regular behaviour at $z=\infty$.

For completeness I would like to mention that I am interested in the Green's functions on 2-dimensional Hyperbolic spaces and was following the book by Borthwick Spectral theory of Infinite-area hyperbolic surfaces, chapter 4. In equation 4.9 he chooses $Q_{\lambda }(z)$ solution in the Legendre equation by demanding the regularity at $z=\infty$. Now, however, if I keep the imaginary part, then the infinity behaviour of the function isn't good and it is not clear why such a choice should be made.

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