'The mother of all continued fractions'

Volume 84 / 2000

Karma Dajani, Cor Kraaikamp Colloquium Mathematicum 84 (2000), 109-123 DOI: 10.4064/cm-84/85-1-109-123


We give the relationship between regular continued fractions and 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 fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full continued fraction), and obtain the ergodic system underlying the Farey expansion.


  • Karma Dajani
  • Cor Kraaikamp

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