Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relations between moments for the Jacobi and Cauchy random matrix
  ensembles   We outline a relation between the densities for the $\beta$-ensembles with
respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval
$(-1,1)$ and the Cauchy weight
$(1-\mathrm{i}x)^{\eta}(1+\mathrm{i}x)^{\bar{\eta}}$ by appropriate analytic
continuation. This has the consequence of implying that the latter density
satisfies a linear differential equation of degree three for $\beta=2$, and of
degree five for $\beta=1$ and $4$, analogues of which are already known for the
Jacobi weight $x^a(1-x)^b$ supported on $(0,1)$. We concentrate on the case
$a=b$ (Jacobi weight on $(-1,1)$) and $\eta$ real (Cauchy weight) since the
density is then an even function and the differential equations simplify. From
the differential equations, recurrences can be obtained for the moments of the
Jacobi weight supported on $(-1,1)$ and/or the moments of the Cauchy weight.
Particular attention is paid to the case $\beta=2$ and the Jacobi weight on
$(-1,1)$ in the symmetric case $a=b$, which in keeping with a recent result
obtained by Assiotis et al.~for the $\beta=2$ case of the symmetric Cauchy
weight (parameter $\eta$ real), allows for an explicit solution of the
recurrence in terms of particular continuous Hahn polynomials. Also for the
symmetric Cauchy weight with $\eta=-\beta(N-1)/2-1-\alpha$, after appropriately
scaling $\alpha$ proportional to $N$, we use differential equations to compute
terms in the $1/N^2$ ($1/N$) expansion of the resolvent for $\beta=2$
($\beta=1,4$).

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