Some Propositions and Lemmas

where , and is a fractional linear transformation.

Since the partial quotients that come from a surd are almost palindromic, it is possible to produce more explicit forms of Proposition 1. The following lemmas reduce the amount of calculation necessary to verify that a set of partial quotients come from a surd by half. They will be used in the proof of the theorems in Section . The rest of the report contains new ideas and results, which are discussed in [5].

Now that these two lemmas have been stated, it is convenient to
introduce and prove the theorems regarding two families of
continued fractions.