^{1, 2}

^{3}

^{1}

^{2}

^{3}

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

In this paper, we consider a well-known property of the Fibonacci sequence, defined by

Many proofs already exist and are well known since long time, and we do not wish to add one more to the repertory.

The Fibonacci sequence can be studied in the framework of the Difference Equations Theory (e.g., see [

The question is:

In the case of the Fibonacci sequence, it is clear that the Golden ratio is the root of the characteristic polynomial with maximum modulus, and all the proofs of (

Surprisingly, the correct answer to the question is

Another surprise: the existence of the dominant root is independent of the existence of the Kepler limit, and also the value of the limit is not necessarily the dominant root. This is shown by the following examples. Consider the sequence

In this case, the dominant root exists, while the Kepler limit does not. In the case

Here, the characteristic polynomial has

These phenomena attracted several researchers since Poincaré, who proved in a more general context (e.g., see [

Let

The condition about the pairwise distinct moduli is optimal (namely, necessary and sufficient) if one requires the existence of the Kepler limit for all possible initial conditions; in other words, if there exist two distinct roots with the same modulus, then it is always possible to consider initial conditions such that the Kepler limit does not exist.

What can be said about the general case, also when the moduli of the roots are not necessarily distinct? Is there a necessary and sufficient condition for the existence of the Kepler limit?

Well, for a given linear recurrence with constant coefficients

In this representation, one can see only a

In [

Thus, Theorem

On the other hand, for a general order-

In the end, the notion of

The importance of this study relies upon the fact that the ratio of consecutive terms tells that all linear recurrences in agreement behave at infinity like geometric sequences, and the first terms (which give the initial conditions), along with the law (which can be conjectured after a reasonable number of tests), permit to predict the ratio. This may be very important for all applications, where linear recurrences represent mathematical models.