  We consider two important families of BC_n-symmetric polynomials, namely
Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials.
We give a family of difference equations satisfied by the former, as well as
generalizations of the branching rule and Pieri identity, leading to a number
of multivariate q-analogues of classical hypergeometric transformations. For
the latter, we give new proofs of Macdonald's conjectures, as well as new
identities, including an inverse binomial formula and several branching rule
and connection coefficient identities. We also derive families of ordinary
symmetric functions that reduce to the interpolation and Koornwinder
polynomials upon appropriate specialization. As an application, we consider a
number of new integral conjectures associated to classical symmetric spaces.
