On the Bauer-Muir transformation for continued fractions and its applications
The Bauer-Muir transformation transforms a continued fraction \(b_ 0+K(a_ n/b_ n)\) into a new continued fraction \(d_ 0+K(c_ n/d_ n)\) with approximants \[ d_ 0+{c_ 1\over d_ 1}+{c_ 2\over d_ 2}+\ldots +{c_ n\over d_ n} =b_ 0+{a_ 1\over b_ 1}+{a_ 2\over b_ 2}+\ldots +{a_ n\over b_ n+w_ n} \] for \(n=0,1,2,..\). for a given sequence \(\{w_ n\}\). Perron made extensive use of this transformation in his classical book on continued fractions to find the value of the continued fraction in some particular cases. His method was based on a theorem which required that all \(a_ n\), \(b_ n\) and \(w_ n\) were positive. This condition is not necessary. In the present paper Perron's theorem is generalized. We also show how the transformation can be used to derive ``better'' continued fractions. In addition, a theorem on convergence of continued fractions with elements polynomials in n is proved. Examples are taken from Ramanujan's work on continued fractions.
scientific article; zbMATH DE number 4163202
scientific article