Page 1 Next

Displaying 1 – 20 of 169

Showing per page

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...

5-dissections and sign patterns of Ramanujan's parameter and its companion

Shane Chern, Dazhao Tang (2021)

Czechoslovak Mathematical Journal

In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction R ( q ) and its reciprocal. We obtain the 5-dissections for functions R ( q ) R ( q 2 ) 2 and R ( q ) 2 / R ( q 2 ) , which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.

A majorant problem.

Peretz, Ronen (1992)

International Journal of Mathematics and Mathematical Sciences

Currently displaying 1 – 20 of 169

Page 1 Next