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ABSTRACT. 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*(£/) •

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The very well-poised elliptic Macdonald functions W λ/µ in n independent variables are defined and their properties are investigated. The W λ/µ are generalized by introducing an extra parameter to the elliptic Jackson coefficients ω λ/µ and their properties are studied. BCn multivariable Jackson sums in terms of both W λ and ω λ functions are proved.

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