Two-sided continued fractions in Banach algebras— A Śleszyński–Pringsheim-type convergence criterion and applications

We consider two-sided continued fractions in Banach algebras, that is K=b0+a1(b1+a2(b2+…)−1c2)−1c1,K=b0+a1(b1+a2(b2+…)−1c2)−1c1, where the coefficients bn,an,cnbn,an,cn are elements of some Banach algebra. We prove convergence criteria which are exact generalizations of the well-known Pringsheim-type convergence criteria for complex continued fractions, and state a result concerning the rate of convergence. Both convergence criteria and error bounds improve all known results.