'The mother of all continued fractions'

In this paper we give the relationship between the regular continued fraction and the Lehner fractions using a procedure known as insertion Starting from the regular continued fraction expansion of any real irrational x when the maximal number of insertions is applied one obtains the Lehner fraction of x Insertions and singularizations show how these and other continued fractions expansions are related We will also investigate the relation between the Lehner fractions and the Farey expansion and obtain the ergodic system underlying the Farey expansio