Properties of the Nonextensive Gaussian entropy

The present work investigates the Lesche stability (experimental robustness), the thermodynamic stability, the Legendre structure of thermodynamics, and derives the Maximum Entropy distribution of the one--parametric ``nonextensive Gaussian'' entropy. We show that this entropy definition fulfills both stability conditions for all values of its parameter ($q\in\mathbb{R}$). The entropy maximizer contains the Lambert $W$--function, which allows the preservation of the Legendre transformations.