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

• Main Author: Hendrik Baumann
• Format: Article
• Language: English
• Published: SpringerOpen 2019-09-01
• Series: Advances in Difference Equations
• Subjects: Generalized continued fractions; Continued fractions in Banach algebras; Matrix continued fractions; Convergence criteria; Pringsheim-type convergence criterion
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2340-9

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.