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.