  Moment evaluations are important for the study of non-classical orthogonal
polynomial systems for which explicit representations are not known. In this
paper we compute, in terms of the hypergeometric function, the moments
associated with a generalized ultraspherical weight on a collection of
intervals with two symmetric gaps. These moments, parametrized by the endpoints
of the gaps, are identified as a one parameter deformation between the full
range ultraspherical moments and the half range ultraspherical moments.
