3x+1 inverse orbit generating functions almost always have natural boundaries
The 3x+k function sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of . We consider the generating functions , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions 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...