On the $k$-generalized Fibonacci numbers with negative indices

In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ 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 \cite{BK} we prove that there are only finitely many perfect powers in $(F_n^{(k)})_{n\in \Z}$, 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 $(n,m)\in \Z^2$. In the case $k=l=4$ we establish all solutions of this equation.