Chain sequence (continued fraction)

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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

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 ≤ 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

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 {an} are a chain sequence.

[edit] 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
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 {¼, ¼, ¼, ...}.

[edit] Notes

  1. ^ Wall traces this result back to Oskar Perron (Wall, 1948, p. 48).

[edit] 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