How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called? Consider a family of distributions with PDF  (up to a proportionality constant) given by
$$p(x)\sim \frac{1}{(1+\alpha x^2)^{1/\alpha}}.$$
How is it called? If it does not have a name, how would you call it?
It looks quite similar to the family of $t$-distributions with PDF proportional to
$$p(x)\sim \frac{1}{(1+\frac{1}{\nu} x^2)^{(\nu+1)/2}}.$$
When $\alpha=\nu=1$ we have $t$-distribution with 1 df, aka Cauchy distribution. When $\alpha\to 0$ or $\nu\to\infty$, we get Gaussian distribution. 
This family of distributions appears in Yang et al., Heavy-Tailed Symmetric Stochastic Neighbor Embedding, NIPS 2009, but they don't use any name to refer to it.
 A: It's simply a particular scaled $t$-distribution -- a $t$-distribution with a different variance to the standard $t$-distribution.
Let $\nu = \frac{2}{\alpha}-1$. Let $\sigma= \frac{\sqrt{2-\alpha}}{\alpha}$. 
Then (if I did it right) $Y=X/\sigma$ is a standard $t$ with $\nu$ d.f.

Here's how my reasoning went:
$$f_Y(y)= c\cdot \frac{1}{(1+\frac{y^2}{\nu})^{(\nu+1)/2}}$$ is a standard $t$-density.
We get the scale family by letting $X/\sigma=Y$, in which case $$f_X(x) = \frac{1}{\sigma}f_Y\Big(\frac{x}{\sigma}\Big) = \frac{c}{\sigma}\cdot \frac{1}{(1+\frac{x^2}{\sigma^2\nu})^{(\nu+1)/2}}$$ is a scaled $t$-density.
Just equate the coefficients in your density to this, and solve for $\nu$ and $\sigma$.
Recognizing that a scale parameter will take up whatever isn't "right" in $\alpha x^2$ (given that $\nu$ is already defined by equating powers) was all that was needed to see it's scaled $t$; algebra wasn't required until it came time to actually find the parameters of the $t$.

[Final note: In case it's not obvious that a scale family has the form $f_X(x) = \frac{1}{\sigma}f_Y\Big(\frac{x}{\sigma}\Big)$, take the probability statement $F_X(x) = F_Y\Big(\frac{x}{\sigma}\Big)$ (noting that the event $X/\sigma\leq t$ is identical to the event $Y\leq t$) and differentiate.]
