I have been trying to understand $q$-exponential family of distributions. In this paper, it has the definition of $q$-logarithm, $q$-exponential and $q$-Gaussian as follows. $$\log_{q}(u) = \frac{1}{1-q}(u^{(1-q)}-1), \ \ \ \exp_{q}(u)=\{1+(1-q)u\}^{\frac{1}{1-q}} \\p(x,\mu,\sigma)=\exp_{q}\big[-\frac{(x-\mu)^{2}}{2\sigma^{2}}-\psi(\mu,\sigma)\big],$$ where $\psi$ is determined by normalization, i.e., $\int p dx = 1$. In wikipedia and some other papers, as a definition of $q$-Gaussian, they have $$p(x) = \frac{\sqrt{\beta}}{C_{q}}\exp_{q}(-\beta x^{2}).$$ Both $p$ s have quadratic term of $x$ inside of $\exp_{e}$, so I can see that they are generalization of Gaussian distribution. I have two questions about these definitions.
If the two representations are equivalent, where $\psi$ in the first(paper) expression goes in the second (wikipedia) expression. How can $\psi$ be pulled out of $\exp_{u}$?
Is it fine to claim that $p$ is $q$-Gaussian if it is written in terms of quadratic function of $x$? What is $\mu$ and $\sigma$ in this case?
Any comments including references would be appreciated. Thank you.