A VARIANT OF THE TYPE-1 BETA AND DIRICHLET INTEGRALS

A double integral, which can be taken as an extension of a type-1 beta integral, is introduced and its properties are studied. Than a k-variate multiple integral on a unit cube in Euclidean k-space is introduced, which will produce the total integral equivalent to the total integral in a (k-1)-variate type-1 Dirichlet integral, or the integral over a simplex in a (k - 1)- at. Several properties of this multiple integral are studied. Marginal functions and Mellin transform are examined. Several transformations are given, nally leading to a type-1 Dirichlet integral. Statistical densities connected with the integrand of the extended Dirichlet integral are also discussed.