A Nonlinear Recursive Sequence

[Maury Barbato]

Hello,
let a, b and c three real numbers, and define

c_0 = c,

c_{n+1} = a + b/c_n.

Did someone study the properties of this sequence?
In particular, does someone know for what values
of a, b, and c the sequence is defined for every
natural number n?
For what values of a,b, and c is it monotonic?
And finally, for what values of a,b, and c does it
converge?

Thank you very very much for your attention.
My Best Regards,
Maurizio Barbato

[Alois Steindl]

Maury Barbato <maurizi...@aruba.it> writes:

Hello,
what have you tried so far?
Why do you need these answers?

Alois

[Maury Barbato]

Alois Steindl wrote:

Ok, I don't really need these answers. While I was
studying a problem in statistics I met a sequence of
this type, and I could prove that it is convergent.
Anyhow, I was quite puzzled from the fact that
studying the general case (that is the sequence
I defined in my post with generic a, b, and c) was much
more complicated (I really don't have any idea of the
answers).

Thank you, Alois, for your interest.

[Robert Israel]

Maury Barbato <maurizi...@aruba.it> writes:

f(z) = a + b/z = (a z + b)/z is a fractional linear transformation,
and so are its iterates: f^[n](z) = (a_n z + b_n)/(A_n z + B_n)
where
[ a_n b_n ]
[ A_n B_n ]
is the n'th power of the matrix
[ a b ]
[ 1 0 ]
Diagonalize this matrix...
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Maury Barbato]

Robert Israel wrote:

Dear Professor Israel, I am very surprised from
the fact that the isomorphism between linear
fractional transformsations and matrices ...
my mathematical ignorance is rellay infinite!
Anyhow, I didn't undertstand exactly your suggestion.
You say, suppose that the matrix

A = [a b]
[1 0]

is diagonalizable, and write

A = M^(-1) * D * M,

where D is diagonal, so that

A^n = M^(-1) * D^n * M,

and

f_n = g^(-1) ° h ° g,

where g is the fractional tranformation corresponding
to M , and h that corresponding to D.
But, I can't see how this helps to study our sequence.
Could you you cast some light on my blind mind?

Thank you very very ... much, prof!
Friendly Regards,
and my sincere wishes of a Happy New Year.
Maurizio Barbato