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

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

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...

A majorant problem.

Peretz, Ronen (1992)

International Journal of Mathematics and Mathematical Sciences

Approximation of entire functions of slow growth on compact sets

G. S. Srivastava, Susheel Kumar (2009)

Archivum Mathematicum

In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized order and generalized type of entire functions of slow growth have been obtained in terms of approximation and interpolation errors.

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