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Convergence of Continued Fractions

  • William B. Jones (a1) and W. J. Thron (a1)

Extract

Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form

1.1

and let

1.2

The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).

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References

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1. Hillam, K. L. and Thron, W. J., A general convergence criterion for continued fractions K(an/bn), Proc. Amer. Math. Soc. 16 (1965), 12561262.
2. Jones, William B. and Thron, W. J., Further properties of T-fractionst Math. Ann. 166 (1966), 106118.
3. Perron, Oskar, Die Lehre von den Kettenbriichen, 3rd éd., Vol. 2 (Teubner, Verlagsgesellschaft, Stuttgart, 1957).
4. Thron, W. J., Convergence regions for the general continued fraction, Bull. Amer. Math. Soc. 49 (1943), 913916.
5. Thron, W. J., Some properties of continued fractions 1 + d0z + K(z/(1 + dnz)), Bull. Amer. Math. Soc. 54 (1948), 206218.
6. Thron, W. J., Convergence of sequences of linear fractional transformations and of continued fractions, J. Indian Math. Soc. 27 (1963), 103127.
7. Wall, H. S., Analytic theory of continued fractions (Van Nostrand, Princeton, N.J., 1948).
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