# 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

${\displaystyle a_{1}=(1-g_{0})g_{1}\quad a_{2}=(1-g_{1})g_{2}\quad a_{n}=(1-g_{n-1})g_{n}}$

where either (a) 0  gn < 1, or (b) 0 < gn  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

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

converges uniformly on the closed unit disk |z|  1 if the coefficients {an} 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 {gn} = {x}, and x < ½, the resulting sequence {an} will be an endless repetition of a real number y that is less than ¼.
• The choice gn = ½ is not the only set of generators for this particular chain sequence. Notice that setting
${\displaystyle g_{0}=0\quad g_{1}={\textstyle {\frac {1}{4}}}\quad g_{2}={\textstyle {\frac {1}{3}}}\quad g_{3}={\textstyle {\frac {3}{8}}}\;\dots }$
generates the same unending sequence {¼, ¼, ¼, ...}.

## Notes

1. 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