Re: uniqueness of fibonacci number extension

On 05.05.2008 21:00, G. A. Edgar wrote:
In article <fvmuka$h9$1@xxxxxxxxxxxxxxxxx>, Jannick Asmus
<jannick.news@xxxxxx> wrote:

On 05.05.2008 03:30, bo198214 wrote:
We know that there is a closed form for the Fibonacci function
(1) F(n)=F(n-1)+F(n-2), F(0)=0, F(1)=1
given by
F(n)=(r^n+(1-r)^n)/sqrt(5), where r=(1+sqrt(5))/2

Actually this is an extension from the domain of definition being the
natural numbers to real or even complex numbers. The extension is
analytic and satisfies equations (1) for all complex n.
Some correction is needed here: For the definition of r^z, z complex,
there is a (branch) of the logarithm needed which does not exist on the
complex plane (but on every simply connected open subset). So (1) cannot
hold if the is meant in terms of holomorphic functions.

You only need a log of r and of 1-r. So choose such logarithms, once
at the start, and use F(n)=(r^n+(1-r)^n)/sqrt(5) ... No worry about
branches, F(n) is defined for n in the whole complex plane.

Actually, F(n)=(r^n+(1-r)^n)/sqrt(5) isn't the Fibonaccis. Should
be F(n)=(r^n-(1-r)^n)/sqrt(5).

Different choices of these logs will give you different solutions,
answering negatively the question of uniqueness, right?

Thanks for making this a lot clearer. I believe such a function does not
even exist by the following reasoning with brute force:

The logarithms of r are ln(r)+2iPi.Z and of 1-r they are ln(r-1)+iPi+2iPi.Z.

Now let us assume that there is a choice of the logarithms of r and 1-r
such that (1) is satisfied for all complex figures, then in particular
for all positive reals x>2. Writing down relation (1) and sorting by
terms r^x, r^(x-1), r^(x-2) and (r-1)^x,... shows that this is
impossible unless exp(iPi.x) has real values only. Contradiction.

My question:
Are there research results which give conditions that makes this
extension unique under all analytic extensions that also satisfy
equations (1) for all complex n? Or is this extension already unique
be these requirements?

--
Best wishes,
J.

.

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