Hurwitz sequences, the Farey process, and general continued fractions
The author studies Hurwitz sequences, obtained by the Farey process, and continued fractions, i.e. fractions \[ [b_ 0;a_ 1/b_ 1,a_ 2/b_ 2,...]=b_ 0+\frac{a_ 1}{b_ 1+}\frac{a_ 2}{b_ 2+}... \] for which \(a_ i\) and \(b_ i\) are integers; \(b_ i>0\) and \(| a_ i| =1\) for \(i=1,2,..\). They have convergents \(P_ n=p_ n/q_ n=[b_ 0;a_ 1/b_ 1,...,a_ n/b_ n].\) Such a unitary continued fraction is a continued fraction for a real number \(\alpha\), if \(P_ n\to \alpha\), as \(n\to \infty.\) The main results of this paper are in connection with the so-called full continued fraction - this is a unitary continued fraction for any irrational \(\alpha\), the sequence of convergents of which contains the sequence of convergents of each other unitary continued fraction of the same \(\alpha\). He proves, for instance, that if \(\alpha\) is a positive irrational number, then there exists a full continued fraction \([b_ 0;a_ 1/b_ 1,...]\) for \(\alpha\), for which \(b_ 0=1\) and \(a_ i/b_ i=1/1\) or - 1/2 for all i and the inverse proposition is also true. \(\{\) Misprint: On page 241 in Definition (2) \((a+c)/(c+d)\) must be read as \((a+c)/(b+d).\}\)
scientific article; zbMATH DE number 4091588
scientific article