First, we write the continued fraction of
in a slightly different
form. Instead of
we can start the sequence of
coefficients
to make the pattern the same throughout. (Everywhere else in
this chapter we assume that the partial quotients
for
are positive, but temporarily relax that
condition here and allow
.) The numerators and
denominators of the convergents given by this new sequence satisfy a
simple recurrence. Using
as a stand-in for
or
, we
have
Our first goal is to collapse these three recurrences into one
recurrence that only makes mention of
,
, and
. We have
This same method of simplification also shows us that
To get rid of
in the first equation, we make the
substitutions
Substituting for
and then
, we finally have the
needed collapsed recurrence,
William
2007-06-01