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 < 1, or (b) 0 < g_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^[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 {a_n} 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 {g_n} = {x}, and x < ½, the resulting sequence {a_n} will be an endless repetition of a real number y that is less than ¼.
The choice g_n = 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, ...}.

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

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