  We present two extended forms of Fisher information that fit well in the
context of nonextensive thermostatistics. We show that there exists an
interplay between these generalized Fisher information, the generalized
$q$-Gaussian distributions and the $q$-entropies. The minimum of the
generalized Fisher information among distributions with a fixed moment, or with
a fixed $q$-entropy is attained, in both cases, by a generalized $q$-Gaussian
distribution. This complements the fact that the $q$-Gaussians maximize the
$q$-entropies subject to a moment constraint, and yields new variational
characterizations of the generalized $q$-Gaussians. We show that the
generalized Fisher information naturally pop up in the expression of the time
derivative of the $q$-entropies, for distributions satisfying a certain
nonlinear heat equation. This result includes as a particular case the
classical de Bruijn identity. Then we study further properties of the
generalized Fisher information and of their minimization. We show that, though
non additive, the generalized Fisher information of a combined system is upper
bounded. In the case of mixing, we show that the generalized Fisher information
is convex for $q\geq1.$ Finally, we show that the minimization of the
generalized Fisher information subject to moment constraints satisfies a
Legendre structure analog to the Legendre structure of thermodynamics.
