On $k$-generalized Fibonacci numbers with negative indices

In these notes, we study $k$-generalized Fibonacci sequences, $(F_n^{(k)})_{n\in {\mathbb Z}}$, with positive and negative indices. Denote by $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even, then the absolute value of the second real root of $T_k(x)$ is minimal among the roots. Combining this with a deep result of Bugeaud and Kaneko [6], we prove that there are only finitely many perfect powers in $(F_n^{(k)})_{n\in {\mathbb Z}}$, provided $k$ is even. Another consequence is that if $k$ and $l$ denote even integers, then the equation $F_m^{(k)} = \pm F_n^{(l)}$ has only finitely many effectively computable solutions in $(n,m)\in {\mathbb Z}^2$. In the case $k=l=4$, we establish all solutions of this equation.