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.
Keywords and Phrases
Extended Dirichlet integral, Mellin transform, power transformations, extended Dirichlet average.
A.M.S. subject classiﬁcation
6B12, 26B15, 33C20, 44A45, 44A05, 60E05, 62E15.