What question did this study set out to answer?

The aim is to establish a convergence theorem relating generalized continued fractions to higher order linear difference equations in Banach algebras.

January 16, 2026Open Access

A Pincherle‐Type Convergence Theorem for Generalized Continued Fractions in Banach Algebras

Key Points

The aim is to establish a convergence theorem relating generalized continued fractions to higher order linear difference equations in Banach algebras.
Deriving a convergence criterion using the adjoint equation of a higher order linear difference equation.
Applying the developed criteria to the Poincaré–Perron difference equation.
Examining selected problems from pure mathematics and applied stochastic processes.
A new Pincherle-type convergence criterion for generalized continued fractions was established.
Showed that certain subdominant solutions can be computed more efficiently with the adjoint equation.

Abstract

This contribution is dedicated to the interdependence of higher order linear difference equations and generalized continued fractions in Banach algebras. It turns out that the computation of certain subdominant solutions of a higher order linear difference equation can be done more efficiently by considering its adjoint equation. The final result is a Pincherle‐type convergence criterion for generalized continued fractions in terms of the adjoint equation. To demonstrate the applicability of the method, it is applied to the Poincaré–Perron difference equation and selected problems from pure mathematics and applied stochastic processes.

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