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Compositions of linear-fractional transformations
V. I. Buslaeva, S. F. Buslaevab a Steklov Mathematical Institute, Russian Academy of Sciences
b Institute of Mathematics, Ukrainian National Academy of Sciences
Abstract:
We study the asymptotic behavior of the compositions $(\mathbf S_n\circ…\circ\mathbf S_1)(z)$ and $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ of linear-fractional transformations $\mathbf S_n(z)$ ($n=1,2,…$) whose fixed points have limits. In particular, if $\mathbf S_n(z)=\alpha_n(\beta_n+z)^{-1}$, then the sequence of compositions $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ at the point $z=0$ coincides with the sequence of convergents of the formal continued fraction
$$
\frac{\alpha_1}{\beta_1+\dfrac{\alpha_2}{\beta_2+\dotsb}}.
$$
The result obtained can be applied in the study of convergence of formal continued fractions.
DOI:
https://doi.org/10.4213/mzm1507
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English version:
Mathematical Notes, 1997, 61:3, 272–277
Bibliographic databases:
UDC:
517.5 Received: 10.11.1996
Citation:
V. I. Buslaev, S. F. Buslaeva, “Compositions of linear-fractional transformations”, Mat. Zametki, 61:3 (1997), 332–338; Math. Notes, 61:3 (1997), 272–277
Citation in format AMSBIB
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