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In the [[complex analysis|analytic theory]] of [[generalized continued fraction|continued fractions]], a '''chain sequence''' is an infinite sequence {''a''<sub>''n''</sub>} of non-negative real numbers chained together with another sequence {''g''<sub>''n''</sub>} of non-negative real numbers by the equations | |||
:<math> | |||
a_1 = (1-g_0)g_1 \quad a_2 = (1-g_1)g_2 \quad a_n = (1-g_{n-1})g_n | |||
</math> | |||
where either (a) 0 ≤ ''g''<sub>''n''</sub> < 1, or (b) 0 < ''g''<sub>''n''</sub> ≤ 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 [[quadratic form|positive definite]] continued fractions. | |||
The infinite continued fraction of [[convergence problem#Worpitzky's theorem|Worpitzky's theorem]] contains a chain sequence. A closely related theorem<ref>[[Hubert Stanley Wall|Wall]] traces this result back to [[Oskar Perron]] (Wall, 1948, p. 48).</ref> shows that | |||
:<math> | |||
f(z) = \cfrac{a_1z}{1 + \cfrac{a_2z}{1 + \cfrac{a_3z}{1 + \cfrac{a_4z}{\ddots}}}} \, | |||
</math> | |||
converges uniformly on the closed unit disk |''z''| ≤ 1 if the coefficients {''a''<sub>''n''</sub>} are a chain sequence. | |||
==An example== | |||
The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting ''g''<sub>0</sub> = ''g''<sub>1</sub> = ''g''<sub>2</sub> = ... = ½, it is clearly a chain sequence. This sequence has two important properties. | |||
*Since ''f''(''x'') = ''x'' − ''x''<sup>2</sup> is a maximum when ''x'' = ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {''g''<sub>''n''</sub>} = {''x''}, and ''x'' < ½, the resulting sequence {''a''<sub>''n''</sub>} will be an endless repetition of a real number ''y'' that is less than ¼. | |||
*The choice ''g''<sub>''n''</sub> = ½ is not the only set of generators for this particular chain sequence. Notice that setting | |||
::<math> | |||
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 | |||
</math> | |||
:generates the same unending sequence {¼, ¼, ¼, ...}. | |||
==Notes== | |||
<references/> | |||
==References== | |||
*H. S. Wall, ''Analytic Theory of Continued Fractions'', D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), ISBN 0-8284-0207-8 | |||
[[Category:Continued fractions]] |
Latest revision as of 09:43, 26 February 2013
In the analytic theory of continued fractions, a chain sequence is an infinite sequence {an} of non-negative real numbers chained together with another sequence {gn} of non-negative real numbers by the equations
where either (a) 0 ≤ gn < 1, or (b) 0 < gn ≤ 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[1] shows that
converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {an} are a chain sequence.
An example
The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = ½, it is clearly a chain sequence. This sequence has two important properties.
- Since f(x) = x − x2 is a maximum when x = ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {gn} = {x}, and x < ½, the resulting sequence {an} will be an endless repetition of a real number y that is less than ¼.
- The choice gn = ½ is not the only set of generators for this particular chain sequence. Notice that setting
- generates the same unending sequence {¼, ¼, ¼, ...}.
Notes
- ↑ Wall traces this result back to Oskar Perron (Wall, 1948, p. 48).
References
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), ISBN 0-8284-0207-8