On the ${q}$-Continuous Natural Exponential Family

In this paper, we present the concept of $q$-natural exponential families within the framework of $q$-calculus, which extends the classical notion by utilizing the $q$-kernel $e_q^{\theta x f(x)^{q-1}}$ in place of the traditional exponential kernel $e^{\theta x}$. This approach allows us to rederive several established results and explore additional parallels with exponential families. The kernel function has been examined in various contexts to evaluate specific integral forms. Inspired by applications in statistical mechanics, we investigate the properties of the $q$-natural exponential family ($q$-NEF). Additionally, we derive the $q$-Laplace transform of a density function and demonstrate the convexity of this function. We establish that the $q$-NEF is uniquely characterized by its $q$-variance function within its $q$-means domain, which is often simpler than the corresponding generating probability measure. This characterization is facilitated by the relationship between the $q$-mean and the $q$-variance function of a $q$-NEF. Moreover, we provide explicit calculations for the $q$-NEFs and $q$-variance functions of the $q$-Gaussian and $q$-exponential distributions.