Q-Gaussian distribution

The $q$-Gaussian distribution has probability density function $$f(x)=\dfrac{\sqrt{\beta}}{C_q} e_q(-\beta x^2),$$ where $e_q$ denotes the Tsallis $q$-exponential and $$C_q = \left\{ \begin{array}{ll} \dfrac{2\sqrt{\pi}\Gamma(\frac{1}{1-q})}{(3-q)\sqrt{1-q}\Gamma(\frac{3-q}{2(1-q)})}&; -\infty < q < 1 \\ \sqrt{\pi} &; q=1 \\ \dfrac{\sqrt{\pi}\Gamma( \frac{3-q}{2(q-1)} )}{\sqrt{q-1}\Gamma(\frac{1}{q-1})} &; 1<q<3. \end{array} \right.$$