Generalized Continued Fraction Expansions With Constant Partial Denominators

Abstract

We study generalized continued fraction expansions of the formwhere N is a fixed positive integer and the partial numerators ai are positive integers for all i. We call these expansions dnN expansions and show that every positive real number has infinitely many dnN expansions for each N . In particular we study the dnN expansions of rational numbers and quadratic irrationals. Finally we show that every positive real number has for each N a dnN expansion with bounded partial numerators.