Separate convergence of general 𝑇-fractions

Abstract

This article is concerned with the separate convergence of the sequences of numerators { A n ( z ) } \{ {A_n}(z)\} and denominators { B n ( z ) } \{ {B_n}(z)\} of the approximants A n ( z ) / ( B n ( z ) {A_n}(z)/({B_n}(z) of the general T {\text {T}} -fraction \[ K\limits _{n = 1}^\infty \left ( {\frac {{{F_n}z}}{{1 + {G_n}z}}} \right ).\] Convergence results for sequences { A n ( z ) / Γ n ( z ) } \{ {A_n}(z)/{\Gamma _n}(z)\} and { B n ( z ) / Γ n ( z ) } \{ {B_n}(z)/{\Gamma _n}(z)\} , where the sequence { Γ n ( z ) } \{ {\Gamma _n}(z)\} is "sufficiently simple" are also derived.