Separate convergence of general 𝑇-fractions

Thron, W. J.

doi:10.1090/S0002-9939-1991-1045151-0

Separate convergence of general $\textrm {T}$-fractions
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by W. J. Thron

Proc. Amer. Math. Soc. 111 (1991), 75-80

DOI: https://doi.org/10.1090/S0002-9939-1991-1045151-0

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Abstract:

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

References

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