Chain sequence

In the analytic theory of continued fractions, a chain sequence is an infinite sequence {a_n} of positive 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.

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