Relations between moments for the Jacobi and Cauchy random matrix ensembles

Abstract

We outline a relation between the densities for the β-ensembles with respect to the Jacobi weight (1 − x)a(1 + x)b supported on the interval (−1, 1) and the Cauchy weight by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for β = 2 and of degree five for β = 1 and 4, analogs of which are already known for the Jacobi weight xa(1 − x)b supported on (0, 1). We concentrate on the case a = b [Jacobi weight on (−1, 1)] and η 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 th..