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 $\beta=2$ ($\beta=1,4$).