Is there any relation between Kummer's function of First kind $\operatorname{M}\left(a,c;x\right)$ and second kind $\operatorname{U}\left(a,c;x\right)$?
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A formula you might find interesting is that it you circle around in the complex plane one, you obtain that $U(a,b; ze^{2\pi i})=e^{-2\pi i b}U(a,b,z)+M(a,b;z)2\pi ie^{-\pi i b}\sin (\pi b m)/\Gamma(1+a-b)\sin(\pi b)$. In particular, the functions can be given in terms of each other if you allow values on different sheets. – Pax Jul 08 '21 at 11:01
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see https://dlmf.nist.gov/13.2 eqn. 13.2.12 – Pax Jul 08 '21 at 11:01