Moments of generalized Cauchy random matrices and continuous-Hahn polynomials

Abstract In this paper we prove that, after an appropriate rescaling, the sum of moments E N ( s stretchy="false">) Tr stretchy="false">| mathvariant="bold">H 2 k + N × Hermitian matrix H sampled according to generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in variable k . This completes picture investigation that began (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for other three classical ensembles random matrices, Gaussian, Laguerre and Jacobi. Our strategy proof somewhat different from one due fact only which has finite number integer moments. arguments also apply, straightforward modifications, cases studied well. We finally obtain differential equation one-point density function eigenvalue distribution establish large asymptotics