A generalization of Selberg’s beta integral

We evaluate several infinite families of multidimensional integrals which are generalizations or analogs of Euler's classical beta integral.We first evaluate a ^-analog of Selberg's beta integral.This integral is then used to prove the Macdonald-Morris conjectures for the affine root systems of types S{Ci) and S{Cj) v and to give a new proof of these conjectures for SiBC;), S{B ( ), 5 , (B / ) V and S*(£/) •