🚀 go-pugleaf

Here is a problem that has been gnawing at me for years.  I am giving up
on it and I'm going to share it with the rest of you.  

Consider the recurrence un = an un-1 + bn un-2 , where we are given
initial values
u0 and u1 and sequences of an's and bn's .  We all know that if pn and
qn are solutions of the recurrence, with p0 = 1, p1 = a1, q0 = 0, q1 =
b1 (and also p-1 = 0, q-1 = 1), then the fractions pn/qn are the
convergents of the continued fraction :

a1+ b1/a2+ b2/a3+... , which may be terminated at any predetermined
point. 

There is also a matrix analogy to this process, namely if we multiply
the 2 X 2 matrices

( ai  1 )                                      (pn  pn-1 )
( bi  0 )  from i = 1 to n, we get the matrix  (qn  qn-1 )  .

This process can be extended, at least with the matrices and the
recurrence.  For
3 dimensions, we would have the recurrence un = an un-1 + bn un-2 + cn
un-3 , with
an, bn and cn given.  If pn, qn and rn are solutions of this recurrence,
with
p-2 = p-1 = 0, p0 = 1 ; q-2 = 0 , q-1 = 1 , q0 = 0 ; r-2 = 1, r-1 = r-0
= 0 
(which implies p1 = a1 , q1 = b1, r1 = c1 ), we can also generate the
sequences
pn, qn and rn as the entries in the following matrix product :

(ai  1  0 )						(pn  pn-1 pn-2 )
(bi  0  1 )  from i = 1 to n , which yields the matrix  (qn  qn-1 qn-2 )
(ci  0  0 )                                             (rn  rn-1 rn-2 )

Now comes the $64 million question:  what objects comparable to
continued fractions
can be conjured from the situation for 3 dimensions?  Same question for
higher dimensions. The extension to such higher dimensions is obvious,
as far as the recurrence and the matrix product are concerned.  But not
so with the continued fraction algorithm.
The difficulty seems to be that a fraction is inherently a binary
operation operating
on two integers, while in the case of 3 or higher dimensions as
indicated above, there
does not seem to be a "ternary" (or higher) process involving an
extension of such
fractions.  

Any and all assistance with this research problem would be appreciated. 
I am receptive
to co-authoring a paper, if this should lead to a productive result. 
Thanks for your
time.  Paul Bruckman