J. Aust. Math. Soc.  77 (2004), 305-319
The unreasonable effectualness of continued function expansions

 Greg Martin   Department of Mathematics   University of British Columbia   Room 121, 1984 Mathematics Road   Vancouver, BC V6T 1Z2   Canada  gerg@math.ubc.ca

Abstract
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as or have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.