Chain sequence

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 &le; gn < 1, or (b) 0 < gn &le; 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| &le; 1 if the coefficients {an} are a chain sequence.

An example
The sequence {$1⁄4$, $1⁄4$, $1⁄4$, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ...                         = $1⁄2$, it is clearly a chain sequence. This sequence has two important properties.


 * Since f(x) = x &minus; x2 is a maximum when x = $1⁄2$, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {gn} = {x}, and x < $1⁄2$, the resulting sequence {an} will be an endless repetition of a real number y that is less than $1⁄4$.
 * The choice gn = $1⁄2$ 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 {$1⁄4$, $1⁄4$, $1⁄4$, ...}.