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

$a_1=(1-g_0)g_1{a}_2=(1-g_1)g_2{a}_n=(1-g_{n-1})g_n$

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

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

converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {a_n} are a chain sequence.

An example

The sequence {Template:Sfrac, Template:Sfrac, Template:Sfrac, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g₀ = g₁ = g₂ = ... = Template:Sfrac, it is clearly a chain sequence. This sequence has two important properties.

• Since f(x) = x − x² is a maximum when x = Template:Sfrac, 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 < Template:Sfrac, the resulting sequence {a_n} will be an endless repetition of a real number y that is less than Template:Sfrac.
• The choice g_n = Template:Sfrac is not the only set of generators for this particular chain sequence. Notice that setting

$g_0=0g_1=\frac{1}{4}{g}_2=\frac{1}{3}{g}_3=\frac{3}{8}\dots$

generates the same unending sequence {Template:Sfrac, Template:Sfrac, Template:Sfrac, ...}.

References

• H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), Template:ISBN