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Chain sequence

In the analytic theory of continued fractions, a chain sequence is an infinite sequence {a**n} of non-negative real numbers chained together with another sequence {g**n} of non-negative real numbers by the equations

: a_1 = (1-g_0)g_1 \quad a_2 = (1-g_1)g_2 \quad a_n = (1-g_{n-1})g_n

where either (a) 0 ≤ g**n n ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite continued fractions.

The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem shows that

: f(z) = \cfrac{a_1z}{1 + \cfrac{a_2z}{1 + \cfrac{a_3z}{1 + \cfrac{a_4z}{\ddots}}}} ,

converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {a**n} are a chain sequence.

An example

The sequence , , , ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = , it is clearly a chain sequence. This sequence has two important properties.

  • Since f(x) = x − x2 is a maximum when x = , this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {g**n} = {x}, and x n} will be an endless repetition of a real number y that is less than .
  • The choice g**n = is not the only set of generators for this particular chain sequence. Notice that setting

:: g_0 = 0 \quad g_1 = {\textstyle\frac{1}{4}} \quad g_2 = {\textstyle\frac{1}{3}} \quad g_3 = {\textstyle\frac{3}{8}} ;\dots

:generates the same unending sequence , , , ...}.

Notes

References

  • H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973),

References

  1. [[Hubert Stanley Wall. Wall]] traces this result back to [[Oskar Perron]] (Wall, 1948, p. 48).
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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