On the existence of ratio limits of weighted $n$-generalized Fibonacci sequences with arbitrary initial conditions

TL;DR

This paper investigates the ratio limits of weighted n-generalized Fibonacci sequences with arbitrary initial conditions, proving that under certain conditions, the ratio converges to a specific zero of the characteristic polynomial.

Contribution

It establishes the existence and value of ratio limits for weighted n-generalized Fibonacci sequences with arbitrary initial conditions under the asymptotic simplicity condition.

Findings

Ratio limits exist for sequences with asymptotically simple characteristic polynomials.

The ratio limit equals the unique zero of the characteristic polynomial.

Results hold for sequences generated from arbitrary nontrivial initial conditions.

Abstract

We study ratio limits of the consecutive terms of weighted $n$-generalized Fibonacci sequences generated from arbitrary complex initial conditions by linear recurrences with arbitrary complex weights. We prove that if the characteristic polynomial of such a linear recurrence is asymptotically simple, then the ratio limit exists for any sequence generated from arbitrary nontrivial initial conditions and it is equal to the unique zero of the characteristic polynomial.