### 3x+1 inverse orbit generating functions almost always have natural boundaries

The 3x+k function ${T}_{k}\left(n\right)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map ${T}_{k}\left(\xb7\right)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of ${T}_{k}\left(\xb7\right)$. We consider the generating functions ${f}_{k,m}\left(z\right)={\sum}_{n>0,n\to m}{z}^{n}$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions ${f}_{k,m}\left(z\right)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...