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

• Autorzy: Jason P. Bell , Jeffrey C. Lagarias
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 30B10: Power series (including lacunary series)
• 26A18: Iteration
• 11B83: Special sequences and polynomials
• 30B40: Analytic continuation
• 37A45: Relations with number theory and harmonic analysis
• 11K31: Special sequences

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 170
• Numer: 2
• Strony: 101-120

Daty

wydano

2015

Twórcy

autor

Jason P. Bell

• Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada

autor

Jeffrey C. Lagarias

• Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A.

Identyfikatory

DOI

10.4064/aa170-2-1