Generalized continued fractions: a unified definition and a Pringsheim-type convergence criterion

Abstract

Abstract In the literature, many generalizations of continued fractions have been introduced, and for each of them, convergence results have been proved. In this paper, we suggest a definition of generalized continued fractions which covers a great variety of former generalizations as special cases. As a starting point for a convergence theory, we prove a Pringsheim-type convergence criterion which includes criteria for the aforementioned special cases. Furthermore, we address several fields in which our definition may be applied.

Keywords

• Generalized continued fractions
• Continued fractions in Banach algebras
• Matrix continued fractions
• Convergence criteria
• Pringsheim-type convergence criterion