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

Jason P. Bell; Jeffrey C. Lagarias

Acta Arithmetica (2015)

- Volume: 170, Issue: 2, page 101-120
- ISSN: 0065-1036

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topJason P. Bell, and Jeffrey C. Lagarias. "3x+1 inverse orbit generating functions almost always have natural boundaries." Acta Arithmetica 170.2 (2015): 101-120. <http://eudml.org/doc/279152>.

@article{JasonP2015,

abstract = {The 3x+k function $T_\{k\}(n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_k(·)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_k(·)$. We consider the generating functions $f_\{k,m\}(z) = ∑_\{n>0, n → m\} z^\{n\}$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_\{k,m\}(z)$ 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 ≥1 to show that $f_\{1,m\}(z)$ 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 $f_\{1, m\}(z)$ is a rational function of z for the remaining values m = 1,2,4,8.},

author = {Jason P. Bell, Jeffrey C. Lagarias},

journal = {Acta Arithmetica},

keywords = {Collatz problem; Skolem-Mahler-Lech theorem; difference equations; discrete dynamical systems},

language = {eng},

number = {2},

pages = {101-120},

title = {3x+1 inverse orbit generating functions almost always have natural boundaries},

url = {http://eudml.org/doc/279152},

volume = {170},

year = {2015},

}

TY - JOUR

AU - Jason P. Bell

AU - Jeffrey C. Lagarias

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

JO - Acta Arithmetica

PY - 2015

VL - 170

IS - 2

SP - 101

EP - 120

AB - The 3x+k function $T_{k}(n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_k(·)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_k(·)$. We consider the generating functions $f_{k,m}(z) = ∑_{n>0, n → m} z^{n}$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k,m}(z)$ 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 ≥1 to show that $f_{1,m}(z)$ 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 $f_{1, m}(z)$ is a rational function of z for the remaining values m = 1,2,4,8.

LA - eng

KW - Collatz problem; Skolem-Mahler-Lech theorem; difference equations; discrete dynamical systems

UR - http://eudml.org/doc/279152

ER -

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