Formal equivalence between Tsallis and extended Boltzmann-Gibbs statistics

A formal correspondence between the q-distribution obtained from the Tsallis entropy and non-Maxwellian distributions obtained from the Boltzmann-Gibbs (BG) entropy is obtained. This formal correspondence is obtained by imposing an infinite number of constraints when one maximizes the BG entropy. Different from the approach of Tsallis, Prato and Plastino [C. Tsallis, D. Prato, A.R. Plastino, Astrophys. Space Sci., 290 (2004) 259-274], we relate the constraints to the central moments, providing a natural meaning to the q-parameter.