T-criterion

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 shows that

$f(z)={\cfrac {a_{1}z}{1+{\cfrac {a_{2}z}{1+{\cfrac {a_{3}z}{1+{\cfrac {a_{4}z}{\ddots }}}}}}}}\,$

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, ...}.

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