The 3x+k function Tkβ(n) sends n to (3n+k)/2 resp. n/2, according
as n is odd, resp. even, where kβ‘Β±1Β (mod6). The map
Tkβ(β ) sends integers to integers, and for mβ₯1 let nβm mean that m is in the forward orbit of n under iteration of
Tkβ(β ). We consider the generating functions fk,mβ(z)=βn>0,nβmβzn, which are holomorphic in the unit disk. We give
sufficient conditions on (k,m) for the functions fk,mβ(z) have the unit
circle {β£zβ£=1} as a natural boundary to analytic continuation. For the
3x+1 function these conditions hold for all mβ₯1 to show that
f1,mβ(z) has the unit circle as a natural boundary except possibly for m=1,2,4 and 8. The 3x+1 Conjecture is equivalent to the assertion that
f1,mβ(z) is a rational function of z for the remaining values m=1,2,4,8.Comment: 15 page