Suppose we have a random matrix variable with the S-Transform below with $n>=1$
$$g(z)=\frac{1}{(1+z)^n}$$
What is the eigenvalue density around 0?
For $n=1$, this corresponds to Marchenko-Pastur law, hence for small eigenvalues $x$ we have
$$f(x)\sim\frac{1}{\pi \sqrt{x}}$$
How do I get the corresponding expression for $n>1$?
Explicit formula of the density is given in 3.61 of Ipsen's thesis but it's complex, I'm wondering if asymptotics can be determined directly from the S-transform.