SOME Q-BETA AND MELLIN-BARNES INTEGRALS ON COMPACT LIE-GROUPS AND LIE-ALGEBRAS

Multidimensional generalizations of beta type integrals of Barnes, Ram anujan, Askey-Wilson, and others are evaluated, These integrals are an alogues of the summation theorems for multilateral hypergeometric seri es associated to the simple Lie algebras of classical type and type G( 2). Many of these integrals can also be written as group integrals ove r a compact Lie group or conjugation invariant integrals over the corr esponding Lie algebra.