On Selberg’s beta integrals

Abstract

Askey and Richards (1989) evaluate Selberg’s first and second beta integrals using Aomoto’s (1987) formidable methodology of setting and solving a first order difference equation. Using this methodology they evaluate certain other beta and gamma type integrals. However, Selberg’s first and second beta and gamma type integrals very elegantly fit within the framework of hypercomplex multivariate normal distribution theory developed by Kabe (1984), and hence can be evaluated using the known multivariate normal distribution theory integrals.